DIFFERENTIAL EQUATIONS

Size: px
Start display at page:

Download "DIFFERENTIAL EQUATIONS"

Transcription

1 Mr. Isaac Akpor Adjei (MSc. Mathematics, MSc. Biostats) April 7, 2017

2 ORDINARY In many physical situation, equation arise which involve differential coefficients. For example: 1 The rate of decrease of temperature of a hot bo. 2 A bo falling freely under gravity. 3 The oscillation on the end of a spring. 4 The decay of a radioactive substance. 5 The decrease in concentration of chemical compound in a reaction.

3 FORMATION OF Differential equations arise or may be derived in a variety of ways. In most cases the problem is to find the dependent variable in terms of the independent one. Eg. Finding x in terms of t or y in terms of x. Differential equations may be formed by direct differentiation. Eg. y = x 3 + 7x 2 + 3x + 7 Eqn 1 = 3x x + 3 Eqn 2 d 2 y 2 = 6x + 14 Eqn 3 d 3 y 3 = 6 Eqn 4

4 Definition Equations which contain an independent variable, a dependent variable and at least one of their derivatives are called DIFFERENTIAL EQUATIONS. Thus a relationship between a variable quantity x and a dependent function y and its derivatives, d 2 y, d 3 y 2 is called an 3 ORDINARY DIFFERENTIAL EQUATION. Equations 2,3 and 4 are differential equations. Other examples of ordinary differential equations 1 = kx 2 x 2 (1 + y) (1 + x)y 2 = 0 3 d 2 y 2 = n 2 y

5 4. ( )2 + 3y = 0 5. x d 3 y + d 2 y 3 + x( 2 )4 = 0 KINDS OF There are two main types of differential equatons. Ordinary Differential Equations(O.D.E. s) These are differential equations with only one independent variables. Partial Differential Equations(P.D.E s) These are differential equations with more than one independent variable.

6 Examples of partial differential equations x z y + y z x = 0 2 f y = 2x 2 x 2 +y 2 2 z x + 2 z 2 y = 0 2 f xx + f yy = 0 2 z x + 2 z 2 y = z 2 t NOTE: This course is concerned more with ordinary differential equations.partial differential equations may be considered slightly.

7 ORDER OF A DIFFERENTIAL EQUATION Differential equations are classified according to the highest derivative which occurs in the equations.thus if the highest derivative that occurs in an equation d n y n,the equation is said to be of order n. On slide 3, we notice that: Equation 2 is of the first order,having only the first derivative. Equation 3 is of the second order. Equation 4 is of the third order.

8 DEGREE OF A DIFFERENTIAL EQUATION The degree of a differential equation is the highest power of the highest derivative which the equation contains. Thus( d 2 y ) = 0 is of the second order and third degree. Considering the examples on slide 4 and 5, we also notice that example 1,2,3 and 5 are of first degree whilst example 4 is of a second degree. NOTE that in example 5 the degree of the equation is determined by the power of the highest derivative d 3 y and not by the fourth power term in Exercises 3 Indicate the order and the degrees of the following ordinary differential equations. 1. ( d 2 y ) = y x 2 2. x + x d 4 y = 3y 4 3.( )4 + xy = x 4. e x = (1 yex )

9 SOLUTIONS OF A DIFFERENTIAL EQUATION We recall that an ordinary differential equation was defined as a relationship between a variable quantity x and a dependent function y and its derivatives. These equations normally arise from physical situations and it is often required to obtain a functional relationship between x and y alone, having eliminated the derivatives. This relation is referred to the SOLUTION of the differential equation. A solution which is COMPLETE or GENERAL must contain a number of arbitrary constants which is equal to the order of the equation. Solutions of the differential equation with the appropriate number of arbitrary constants are called GENERAL SOLUTIONS.

10 In physical problems, solutions are usually required which satisfy certain specified conditions. These provide information from values to be assigned to the arbitrary constants. This type of solution, which satisfies certain definite conditions, is called a PARTICULAR SOLUTION and the conditions satisfied are called BOUNDARY CONDITIONS or INITIAL CONDITIONS.

11 Example Consider = x which is of the first order. Integrating we have = x y = 1 2 x 2 + A Now if we consider the general solutions y = 1 2 x 2 + A to the equation = x. Let us assume that the boundary condition is given to be y = 1 when x = 0 A = 1 The value assigned to A = 1 and the particular solution is y = 1 2 x 2 + 1

12 OF THE FIRST ORDER AND FIRST DEGREE Let us first look at the case where one variable is absent. (a) When y is absent The general form is = f (x) = f (x) y = f (x)

13 Example Solve the differential equation = x 4 + sin x = (x 4 + sin x) y = 1 5 x 5 cos x + c

14 (b) When x is absent The general form is = f (y) = f (y) Rewriting this in the form : = 1 f (y) 1 = f (y) = f (y) Example Solve the equation = tan y

15 Solution = tan y = 1 tan y = cos y sin y cos y sin y = ln sin y +c x = The examples given above lead us to the main types of first order, first degree differential equations and their solutions.

16 TYPES OF FIRST ORDER 1 Variables Separable 2 Homogeneous 3 Linear 4 Exact

17 TYPE 1 - VARIABLES SEPARABLE If the terms of the equation can rearranged into two groups, each containing only one variable, the variables are said to be SEPARABLE. Since the differential equations of the first order and first degree contain to the first power only, they can be written as = F (x, y) In many cases F (x, y) may be written as F (x, y) = f (x)g(y) where f (x) and g(y) are functions of x only and g(y) is a function of y only.

18 We may then separate the variables and write. g(y) = f (x)

19 Worked Example On Type 1 Example Solve = 5x 7y Solution The variables are separable 7y = 5x 7y = 5x 7 y = 5 x y 2 = x 2 + C y 2 = 5 7 x 2 + C Note : C = C 1 + C 2 and C is an arbitrary constant.

20 Exercise 1 Solve x + y = xy(x y) Solution x + y ( = xy x ) y (y x 2 y) = (x + xy 2 ) = x(1 + y 2 ) y 1 + y 2 = x 1 x ln 1 + y 2 = 1 2 ln 1 x 2 + ln C 1 (1 + y 2 ) 1 2 = C1 (1 + x 2 ) 1 2 (1 + y 2 ) = C(1 + x 2 )

21 Exercise 2 Solve Solution Let z = x + y = (x + y)2 dz = 1 + (1) = = z2 (2) From (1) = dz 1 = dz 1 = z2

22 Solution Cont d dz = z2 + 1 dz z = tan 1 z = x + c z = tan(x + c) x + y = tan(x + c) y = tan(x + c) x

23 Solve the following equations 1 i = tan2 (x + y) ii = (x + 4y)2 2 i (1 + x)y + (1 x)y = 0 ii + k x = i (x + 1) y = 0 ii (y 2 x 2 ) + 2xy

24 HOMOGENEOUS TYPE HOMOGENEOUS M(x, y) is said to be a homogeneous function of degree n if the sum of the powers of x and y in each term of M is n Eg. (i) x 2 y 3xy 2 + 2y 3 is homogeneous of degree 3 (ii) x 4 2x 2 y 2 is homogeneous of degree 4. If a first order D.E. is written in the form = M(x, y) N(x, y), where M and N are homogeneous functions of the same degree, then the equation is said to be HOMOGENEOUS.

25 Examples: i = xy x 2 + y 2 ii (x 2 + y 2 ) = xy Check i. (x 2 + y) = xy? ii. = y(3x 2 + y 2 ) x(x + 3y)?

26 METHOD OF FINDING SOLUTION TO A HOMOGENEOUS DIFFERENTIAL EQUATION If, in the equation; = M(x, y) N(x, y), (3) both M and N are homogeneous of degree n, we may divide them by x n and express the R.H.S as a function of the single variable v, where v = y x y = vx = v + x dv (4) Substituting (4) in the differential equation (3), we find that the result is a new differential equation in which the variables v and x and finally replace v by y x.

27 Example 1 (x 2 + y 2 ) = xy Let v = y x y = vx = v + x (1)

28 Now (x 2 + y 2 ) = xy = xy y x 2 + y 2 = x 1 + ( y x )2 = v 1 + v 2 (2) From (1) and (2) v + x dv = v 1 + v 2

29 x dv = v 1 + v 2 v = v v 2 x dv = v v 2 Now we notice that x and v are separable. So we separate the variables, integrate and substitute for v to obtain the general solution. Thus where y x 2 x dv = v v 2 x = ( 1 v 3 1 v )dv ln(x) = 1 ln(v) + ln(a) 2v 2 ln(a) = C

30 y A = e x 2 2y 2 y = Ae x 2 2y 2

31 Example 2 Example 2 = y(3x 2 + y 2 ) x(x 2 + 3y 2 ) Let v = y x y = vx = v + x

32 v + x y = x (3 + ( y x )2 ) 1 + 3( y x )2 v + x = v(3 + v 2 ) 1 + 3v 2 x = 1 + 3v 2 2v(1 v 2 ) dv

33 ln(x) = 1 2 [ 1 v v 2 ] dv 1 + v Finally, we have log e x 2 = log e Av (1 v 2 ) 2 (x 2 y 2 ) 2 = Axy

34 Linear Type of Ordinary Differential Equations If a differential equation can be written in the form + Py = Q, where P and Q are functions of x only, the equation is said to be LINEAR of the first order since and y occurs linearly. Examples: 1 + 2y cot x = cos x 2 + x 1 + x 2 y = 1 2x(1 + x 2 ) 3 x (1 x 2 y = 1 (1 x y tan x = sec x 5 2xy = 2x 6 + xy = x

35 METHODS OF SOLUTION In the standard linear equation + P(y) = Q, the presence of the terms and y suggests the differentiation of a product involving y. To produce this product we multiply the equation throughout by a function u to be determined later. Thus we have u + up(y) = uq

36 The equation reduces to du = up du u = P log e u = P u = e P

37 We note that no arbitrary constants needs to be included here since the constant required in the solution of the original differential equation will arise on performing the integration. The function u = e P is referred to as the INTEGRATING FACTOR (I.F)

38 Example 1 Solve x + 2y = ex Solution x Writing in standard form yields + 2y = ex + 2 x y = ex x ( ) P = 2 x 2 P = = 2 ln(x) = ln(x)2 x

39 Solution Cont d Thus Multiplying (*) by u = x 2, we have u = e P = e 2 x e log e (x)2 = (x) 2 x 2 + 2xy = xex d(x 2 y) = xe x

40 x 2 y = xe x x 2 y = (x 1)e x + c 1 y = ex (x 1) x 2 + c 1 x 2 y = ex (x 1) x 2 + c 2

41 Example 2 Solve the equation cos x + y sin x = 1 Solution Writing in standard form: Now + sin x y = sec x ( ) cos x I.F, u = e sin x cos x = e tan x sin x = ln cos x = ln sec x cos x I.F, u = e ln sec x = ln sec x

42 Solution Multiplying through (*) by the I.F = ln sec x,we have sec x + sin x (sec x)y = sec x sec x cos x sec x + tan x(sec x)y = sec2 x y sec x = sec 2 x = tan x + c 1 y = sin x + c 2 y = tan x sec x + C 1 sec x

43 Exercises Solve the following 1 (x 2 + 1) + 2xy = 4x 2 given that when ( ) x=3, y=4 I.F= (x 2 + 1) Soln: y = 4x 3 x x x(1 x 2 ) + (2x 2 1)y = x 3 (I.F = 1 x ; y=x + A 1 x 1x 1 x 2 = x + A 2 2 ) 3 = y x (y = x ce x ) 4 tan x = 1 + y (y + 1 = c sin x)

44 SOME APPLICATIONS OF We recall that a derivative is a rate of change. It is this idea that gives the differential equation a wide range of applications in the sciences, in the business and social sciences.many of the applications involve a rate of change of some quantity with respect to time. Thus,if the rate of change of y with respect to time, t is proportional to y, then dt y dt = ky The constant k is referred to as the constant of proportionality. Consider a general solution to a differential equation y = ce kt If k is positive, the function represents EXPONENTIAL GROWTH

45 If k is negative, it represents EXPONENTIAL DECAY If t=0,y=c Here c is referred to as the initial value (value of y at time t=0) SOME EXAMPLES AND EXERCISES 1 In a certain type of chemical reaction, the rate at which an old substance (initial amount, a) is converted into a new substance is proportional to both the amount 2 The rate of change in temperature T of a small object placed in a large bo of water with a temperature of 32 C is proportional to the difference between the temperature of the object and the temperature of the water. Find the differential equation that represents this function and find its general solution. 3 Newtons Law of Cooling states that the rate of decrease of temperature of a hot bo is proportional to its excess temperature over that of the surroundings.( dθ dt (θ θ 0)) where θ 0 is the temperature at time t.

46 SOLVED EXAMPLES Example 1 A second order rate chemical reaction is governed by the differential equation dt = k(5 x)2,where x is the change in concentration at time t. x is initially zero and is found to have the value x=1 when t=10s. Find the value of the reaction rate constant k and the values of x when t=20s and t=100s Soln dt = k(5 x)2 = k dt = kt (5 x) 2 (5 x) =? 2 Let u=5-x, du=- (5 x) = du 2 u 2 = 1 5 x + c = u 2 du = u 1 1 = 1 u + c

47 kt = 1 5 x + c when t=0,x=0 c = 1 5 kt = 1 5 x 1 5 when t=10, x=1 10k = = 1 20 k = t = 1 5 x 1 5 when t=20, we have = 1 5 x x = = = x = = 10 3 x = = 5 3 when t=100, we have = 1 5 x x = = 7 10, 5 x = 10 7, x = = 25 7

48 CHEMICAL RATE EQUATIONS ORDER OF A REACTION The order of a chemical reaction is the sum of the powers of the concentration terms that occur in the differential form of the Rate equation. Example: dt = k(a x)0 ZERO ORDER REACTION dt = k(a x) FIRST ORDER REACTION dt = k(a x)2 SECOND ORDER REACTION dt = k(a x)3 THIRD ORDER REACTION where a is initial concentration and x is the decrease in concentration in the chemical reaction.

49 FIRST ORDER REACTION A reaction in which the rate depends on two concentration terms in which one species is present in a very high concentration relative to the other such that its concentration considered to be constant during the course of the reaction results in a first order rate equation dt (a x) dt = k(a x), where a is the initial concentration and x is the decrease in concentration. Example 1. CH 3 CO O C 2 H 5 + H 2 O CH 3 COOH + C 2 H 5 OH dt = [Ester][H 2O] 2.C(CH 3 ) 3 OH C 4 H 8 + H 2 O

50 Solution dt = k(a x) a x = k dt ln(a x) = kdt + c At t=0, x=0 c = ln a ln(a x) = kt ln a ln a a x = kt Eqn1 a a x = ekt Eqn1b SECOND ORDER REACTIONS A + B C A and B are Reactants, C is the product. If x is the decrease in concentration of A at time t, and a and b are the initial concentrations of A and B, then dt = k(a x)(b x) For special cases in which a and b are equimolar amounts.

51 dt = k(a x)2 Thus = k dt (a x) 2 (a x) 1 = kt + C At t=0, x=0, C= 1 a 1 a x = kt + 1 a kt = 1 kt = a x 1 a x a(a x) Eqn2 Now let us consider the solution of the first order rate equation = k(a x) dt Solution: a a x = ekt ln a a x = kt t = 1 k ln a a x

52 Usually ln(a x) is plotted against t k is the slope ln a is the intercept. Note: For Radioisotope work kt = ln No N where N o initial activity and N is the activity at time t. HALF LIFE (t 1 2 ) The half life (t 1 ) is the time taken for half or 50% reaction to occur 2 OR the time taken for the concentration of the reactants to reduce to half (50%) of the initial concentration.

53 Examples FIRST ORDER REACTION kt = a a x t = 1 t ln a a x HALF LIFE At t 1, x = a 2 2 t 1 = 1 2 k ln a a a 2 = 1 k = ln 2 2 k = ln 2a 2a a = 1 k ln 2 t 1 k Thus t 1 does not depend on a 2 t 1 = t 50% = 1 2 k ln a = 1 a 50a k ln a 50a = 1 100a k ln 50a = 1 k ln 2 = k t 90% = 1 k ln = 1 k = k a a 90a 100 ln = 1 k ln 10 = 1 k ln a 10a 100

54 t 80% = 1 k ln = 1 k = k a a 80a 100 ln = 1 k ln 5 = 1 k ln a 20a 100 For the second order reaction, kt = x a(a x) Half life,t 1 when x = a 2 2 kt 1 = a 2 2 a(a a 2 ) t 1 = 1 2 ka Example In a certain thermolecular reaction the decrease x in concentration of a substance R is given by dt = k(a x)3,where k is the reaction rate constant and a is the initial concentration of R. Find the concentration at time t.

55 Solution dt (a x) 3 = k(a x)3 = k dt 1 2(a x) = kt + C 2 At t=0, x=0 1 2(a 0) = k(0) + C C = 1 2 2a 2 Thus 1 2(a x) 2 = kt + 1 2a 2 a2 (a x) = 1 + 2a 2 kt 2 (a x) 2 = a2 1+2a 2 kt a x = a 2 1+2a 2 kt The concentration of the substance R is given as (a x) = a 1+2a2 kt

56 RADIOACTIVE DECAY When radioactive substances decay, the number of atoms that decay in a fixed time period is proportional to the number of atoms at the start of that period. Thus the rate of decay of a radioactive substance is proportional to the number of atoms N present at time t. If the constant of proportionality is λ the decay constant and initially are N O atoms present, then dn dt N dn dt dn N = λn = λ dt ln N = λt + C When t=0, N =N o C = ln N o

57 ln N = λt + ln N o ln N ln N o = λt ln N N o = λt = e λt N N o N = N o e λt Example 1 Carbon 14, one of the three isotopes of carbon is radioactive and decays at a rate which is proportional to the amount present. Its half life i 5570 years. If 10 grams were present originally, how much will be left after 2000 years.

58 Solution dn dt N dn dt dn N = λn = λ dt ln N = λt + C When t=0, N =N o C = ln N o ln N = λt + ln N o ln N ln N o = λt ln N N o = λt = e λt N N o N = N o e λt

59 At half life, N = No 2, t 1 2 = 5570 No 2 = N oe λ(5570) 1 2 = e λ(5570) 5570λ = ln 1 2 λ = ln 1 2 = N = N o e t At t=2000 years,n o = 10grams N = 10e (2000) = grams 7.8 grams.

Differential Equations & Separation of Variables

Differential Equations & Separation of Variables Differential Equations & Separation of Variables SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should reference Chapter 8. of the recommended textbook (or the equivalent

More information

Introduction to Differential Equations

Introduction to Differential Equations Chapter 1 Introduction to Differential Equations 1.1 Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known

More information

First Order Differential Equations Chapter 1

First Order Differential Equations Chapter 1 First Order Differential Equations Chapter 1 Doreen De Leon Department of Mathematics, California State University, Fresno 1 Differential Equations and Mathematical Models Section 1.1 Definitions: An equation

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Chapter 1 Introduction and Basic Terminology Most of the phenomena studied in the sciences and engineering involve processes that change with time. For example, it is well known

More information

Differential Equations

Differential Equations Differential Equations A differential equation (DE) is an equation which involves an unknown function f (x) as well as some of its derivatives. To solve a differential equation means to find the unknown

More information

Solving differential equations (Sect. 7.4) Review: Overview of differential equations.

Solving differential equations (Sect. 7.4) Review: Overview of differential equations. Solving differential equations (Sect. 7.4 Previous class: Overview of differential equations. Exponential growth. Separable differential equations. Review: Overview of differential equations. Definition

More information

Lesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods.

Lesson 3: Linear differential equations of the first order Solve each of the following differential equations by two methods. Lesson 3: Linear differential equations of the first der Solve each of the following differential equations by two methods. Exercise 3.1. Solution. Method 1. It is clear that y + y = 3 e dx = e x is an

More information

9.3: Separable Equations

9.3: Separable Equations 9.3: Separable Equations An equation is separable if one can move terms so that each side of the equation only contains 1 variable. Consider the 1st order equation = F (x, y). dx When F (x, y) = f (x)g(y),

More information

Higher-order ordinary differential equations

Higher-order ordinary differential equations Higher-order ordinary differential equations 1 A linear ODE of general order n has the form a n (x) dn y dx n +a n 1(x) dn 1 y dx n 1 + +a 1(x) dy dx +a 0(x)y = f(x). If f(x) = 0 then the equation is called

More information

3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y:

3 Algebraic Methods. we can differentiate both sides implicitly to obtain a differential equation involving x and y: 3 Algebraic Methods b The first appearance of the equation E Mc 2 in Einstein s handwritten notes. So far, the only general class of differential equations that we know how to solve are directly integrable

More information

Chapter 2. First-Order Differential Equations

Chapter 2. First-Order Differential Equations Chapter 2 First-Order Differential Equations i Let M(x, y) + N(x, y) = 0 Some equations can be written in the form A(x) + B(y) = 0 DEFINITION 2.2. (Separable Equation) A first-order differential equation

More information

Applications of First Order Differential Equation

Applications of First Order Differential Equation Dr Mansoor Alshehri King Saud University MATH204-Differential Equations Center of Excellence in Learning and Teaching 1 / 39 Orthogonal Trajectories How to Find Orthogonal Trajectories Growth and Decay

More information

and verify that it satisfies the differential equation:

and verify that it satisfies the differential equation: MOTIVATION: Chapter One: Basic and Review Why study differential equations? Suppose we know how a certain quantity changes with time (for example, the temperature of coffee in a cup, the number of people

More information

Chapter 6 Differential Equations and Mathematical Modeling. 6.1 Antiderivatives and Slope Fields

Chapter 6 Differential Equations and Mathematical Modeling. 6.1 Antiderivatives and Slope Fields Chapter 6 Differential Equations and Mathematical Modeling 6. Antiderivatives and Slope Fields Def: An equation of the form: = y ln x which contains a derivative is called a Differential Equation. In this

More information

6.5 Separable Differential Equations and Exponential Growth

6.5 Separable Differential Equations and Exponential Growth 6.5 2 6.5 Separable Differential Equations and Exponential Growth The Law of Exponential Change It is well known that when modeling certain quantities, the quantity increases or decreases at a rate proportional

More information

First Order Differential Equations

First Order Differential Equations Chapter 2 First Order Differential Equations Introduction Any first order differential equation can be written as F (x, y, y )=0 by moving all nonzero terms to the left hand side of the equation. Of course,

More information

Chapters 8.1 & 8.2 Practice Problems

Chapters 8.1 & 8.2 Practice Problems EXPECTED SKILLS: Chapters 8.1 & 8. Practice Problems Be able to verify that a given function is a solution to a differential equation. Given a description in words of how some quantity changes in time

More information

Differential Equations: Homework 2

Differential Equations: Homework 2 Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y

More information

6 Second Order Linear Differential Equations

6 Second Order Linear Differential Equations 6 Second Order Linear Differential Equations A differential equation for an unknown function y = f(x) that depends on a variable x is any equation that ties together functions of x with y and its derivatives.

More information

dy dx dx = 7 1 x dx dy = 7 1 x dx e u du = 1 C = 0

dy dx dx = 7 1 x dx dy = 7 1 x dx e u du = 1 C = 0 1. = 6x = 6x = 6 x = 6 x x 2 y = 6 2 + C = 3x2 + C General solution: y = 3x 2 + C 3. = 7 x = 7 1 x = 7 1 x General solution: y = 7 ln x + C. = e.2x = e.2x = e.2x (u =.2x, du =.2) y = e u 1.2 du = 1 e u

More information

Math Applied Differential Equations

Math Applied Differential Equations Math 256 - Applied Differential Equations Notes Existence and Uniqueness The following theorem gives sufficient conditions for the existence and uniqueness of a solution to the IVP for first order nonlinear

More information

APPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai

APPLIED MATHEMATICS. Part 1: Ordinary Differential Equations. Wu-ting Tsai APPLIED MATHEMATICS Part 1: Ordinary Differential Equations Contents 1 First Order Differential Equations 3 1.1 Basic Concepts and Ideas................... 4 1.2 Separable Differential Equations................

More information

4 Differential Equations

4 Differential Equations Advanced Calculus Chapter 4 Differential Equations 65 4 Differential Equations 4.1 Terminology Let U R n, and let y : U R. A differential equation in y is an equation involving y and its (partial) derivatives.

More information

Chapter 6: Messy Integrals

Chapter 6: Messy Integrals Chapter 6: Messy Integrals Review: Solve the following integrals x 4 sec x tan x 0 0 Find the average value of 3 1 x 3 3 Evaluate 4 3 3 ( x 1), then find the area of ( x 1) 4 Section 6.1: Slope Fields

More information

ECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0

ECONOMICS 207 SPRING 2006 LABORATORY EXERCISE 5 KEY. 8 = 10(5x 2) = 9(3x + 8), x 50x 20 = 27x x = 92 x = 4. 8x 2 22x + 15 = 0 (2x 3)(4x 5) = 0 ECONOMICS 07 SPRING 006 LABORATORY EXERCISE 5 KEY Problem. Solve the following equations for x. a 5x 3x + 8 = 9 0 5x 3x + 8 9 8 = 0(5x ) = 9(3x + 8), x 0 3 50x 0 = 7x + 7 3x = 9 x = 4 b 8x x + 5 = 0 8x

More information

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x). You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =

More information

Math 147 Exam II Practice Problems

Math 147 Exam II Practice Problems Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab

More information

Techniques of Integration

Techniques of Integration Chapter 8 Techniques of Integration 8. Trigonometric Integrals Summary (a) Integrals of the form sin m x cos n x. () sin k+ x cos n x = ( cos x) k cos n x (sin x ), then apply the substitution u = cos

More information

6.1 Antiderivatives and Slope Fields Calculus

6.1 Antiderivatives and Slope Fields Calculus 6. Antiderivatives and Slope Fields Calculus 6. ANTIDERIVATIVES AND SLOPE FIELDS Indefinite Integrals In the previous chapter we dealt with definite integrals. Definite integrals had limits of integration.

More information

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) =

Solutions to Exam 1, Math Solution. Because f(x) is one-to-one, we know the inverse function exists. Recall that (f 1 ) (a) = Solutions to Exam, Math 56 The function f(x) e x + x 3 + x is one-to-one (there is no need to check this) What is (f ) ( + e )? Solution Because f(x) is one-to-one, we know the inverse function exists

More information

Basic Theory of Differential Equations

Basic Theory of Differential Equations page 104 104 CHAPTER 1 First-Order Differential Equations 16. The following initial-value problem arises in the analysis of a cable suspended between two fixed points y = 1 a 1 + (y ) 2, y(0) = a, y (0)

More information

Introductory Differential Equations

Introductory Differential Equations Introductory Differential Equations Lecture Notes June 3, 208 Contents Introduction Terminology and Examples 2 Classification of Differential Equations 4 2 First Order ODEs 5 2 Separable ODEs 5 22 First

More information

Math 222 Spring 2013 Exam 3 Review Problem Answers

Math 222 Spring 2013 Exam 3 Review Problem Answers . (a) By the Chain ule, Math Spring 3 Exam 3 eview Problem Answers w s w x x s + w y y s (y xy)() + (xy x )( ) (( s + 4t) (s 3t)( s + 4t)) ((s 3t)( s + 4t) (s 3t) ) 8s 94st + 3t (b) By the Chain ule, w

More information

Section 2.2 Solutions to Separable Equations

Section 2.2 Solutions to Separable Equations Section. Solutions to Separable Equations Key Terms: Separable DE Eponential Equation General Solution Half-life Newton s Law of Cooling Implicit Solution (The epression has independent and dependent variables

More information

FINAL REVIEW FALL 2017

FINAL REVIEW FALL 2017 FINAL REVIEW FALL 7 Solutions to the following problems are found in the notes on my website. Lesson & : Integration by Substitution Ex. Evaluate 3x (x 3 + 6) 6 dx. Ex. Evaluate dt. + 4t Ex 3. Evaluate

More information

Math 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3

Math 201 Solutions to Assignment 1. 2ydy = x 2 dx. y = C 1 3 x3 Math 201 Solutions to Assignment 1 1. Solve the initial value problem: x 2 dx + 2y = 0, y(0) = 2. x 2 dx + 2y = 0, y(0) = 2 2y = x 2 dx y 2 = 1 3 x3 + C y = C 1 3 x3 Notice that y is not defined for some

More information

Math 225 Differential Equations Notes Chapter 1

Math 225 Differential Equations Notes Chapter 1 Math 225 Differential Equations Notes Chapter 1 Michael Muscedere September 9, 2004 1 Introduction 1.1 Background In science and engineering models are used to describe physical phenomena. Often these

More information

Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012

Polytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012 Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet

More information

First Order Differential Equations

First Order Differential Equations Chapter 2 First Order Differential Equations 2.1 9 10 CHAPTER 2. FIRST ORDER DIFFERENTIAL EQUATIONS 2.2 Separable Equations A first order differential equation = f(x, y) is called separable if f(x, y)

More information

Handbook of Ordinary Differential Equations

Handbook of Ordinary Differential Equations Handbook of Ordinary Differential Equations Mark Sullivan July, 28 i Contents Preliminaries. Why bother?...............................2 What s so ordinary about ordinary differential equations?......

More information

Title: Solving Ordinary Differential Equations (ODE s)

Title: Solving Ordinary Differential Equations (ODE s) ... Mathematics Support Centre Title: Solving Ordinary Differential Equations (ODE s) Target: On completion of this workbook you should be able to recognise and apply the appropriate method for solving

More information

Solutions of Math 53 Midterm Exam I

Solutions of Math 53 Midterm Exam I Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior

More information

Series Solution of Linear Ordinary Differential Equations

Series Solution of Linear Ordinary Differential Equations Series Solution of Linear Ordinary Differential Equations Department of Mathematics IIT Guwahati Aim: To study methods for determining series expansions for solutions to linear ODE with variable coefficients.

More information

7.1. Calculus of inverse functions. Text Section 7.1 Exercise:

7.1. Calculus of inverse functions. Text Section 7.1 Exercise: Contents 7. Inverse functions 1 7.1. Calculus of inverse functions 2 7.2. Derivatives of exponential function 4 7.3. Logarithmic function 6 7.4. Derivatives of logarithmic functions 7 7.5. Exponential

More information

Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y

Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y 10 Differential Equations Test Form A 1. Find the general solution to the first order differential equation: y 1 yy 0. 1 (a) (b) ln y 1 y ln y 1 C y y C y 1 C y 1 y C. Find the general solution to the

More information

MathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited

MathsGeeks. Everything You Need to Know A Level Edexcel C4. March 2014 MathsGeeks Copyright 2014 Elite Learning Limited Everything You Need to Know A Level Edexcel C4 March 4 Copyright 4 Elite Learning Limited Page of 4 Further Binomial Expansion: Make sure it starts with a e.g. for ( x) ( x ) then use ( + x) n + nx + n(n

More information

Solutions to Section 1.1

Solutions to Section 1.1 Solutions to Section True-False Review: FALSE A derivative must involve some derivative of the function y f(x), not necessarily the first derivative TRUE The initial conditions accompanying a differential

More information

Chapter1. Ordinary Differential Equations

Chapter1. Ordinary Differential Equations Chapter1. Ordinary Differential Equations In the sciences and engineering, mathematical models are developed to aid in the understanding of physical phenomena. These models often yield an equation that

More information

MATH1013 Calculus I. Derivatives II (Chap. 3) 1

MATH1013 Calculus I. Derivatives II (Chap. 3) 1 MATH1013 Calculus I Derivatives II (Chap. 3) 1 Edmund Y. M. Chiang Department of Mathematics Hong Kong University of Science & Technology October 16, 2013 2013 1 Based on Briggs, Cochran and Gillett: Calculus

More information

Exam 1 Review: Questions and Answers. Part I. Finding solutions of a given differential equation.

Exam 1 Review: Questions and Answers. Part I. Finding solutions of a given differential equation. Exam 1 Review: Questions and Answers Part I. Finding solutions of a given differential equation. 1. Find the real numbers r such that y = e x is a solution of y y 30y = 0. Answer: r = 6, 5 2. Find the

More information

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone

3.4 Conic sections. Such type of curves are called conics, because they arise from different slices through a cone 3.4 Conic sections Next we consider the objects resulting from ax 2 + bxy + cy 2 + + ey + f = 0. Such type of curves are called conics, because they arise from different slices through a cone Circles belong

More information

Today: 5.4 General log and exp functions (continued) Warm up:

Today: 5.4 General log and exp functions (continued) Warm up: Today: 5.4 General log and exp functions (continued) Warm up: log a (x) =ln(x)/ ln(a) d dx log a(x) = 1 ln(a)x 1. Evaluate the following functions. log 5 (25) log 7 p 7 log4 8 log 4 2 2. Di erentiate the

More information

Mathematics for Engineers II. lectures. Differential Equations

Mathematics for Engineers II. lectures. Differential Equations Differential Equations Examples for differential equations Newton s second law for a point mass Consider a particle of mass m subject to net force a F. Newton s second law states that the vector acceleration

More information

EXPONENTIAL, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS

EXPONENTIAL, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS Calculus for the Life Sciences nd Edition Greenwell SOLUTIONS MANUAL Full download at: https://testbankreal.com/download/calculus-for-the-life-sciences-nd-editiongreenwell-solutions-manual-/ Calculus for

More information

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv

Problem 1 (Equations with the dependent variable missing) By means of the substitutions. v = dy dt, dv V Problem 1 (Equations with the dependent variable missing) By means of the substitutions v = dy dt, dv dt = d2 y dt 2 solve the following second-order differential equations 1. t 2 d2 y dt + 2tdy 1 =

More information

Lecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25

Lecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25 Lecture 2 Introduction to Differential Equations Roman Kitsela October 1, 2018 Roman Kitsela Lecture 2 October 1, 2018 1 / 25 Quick announcements URL for the class website: http://www.math.ucsd.edu/~rkitsela/20d/

More information

LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS

LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS 130 LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS: A differential equation (DE) is an equation involving an unknown function and one or more of its derivatives. A differential

More information

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly

More information

= f (x ), recalling the Chain Rule and the fact. dx = f (x )dx and. dx = x y dy dx = x ydy = xdx y dy = x dx. 2 = c

= f (x ), recalling the Chain Rule and the fact. dx = f (x )dx and. dx = x y dy dx = x ydy = xdx y dy = x dx. 2 = c Separable Variables, differential equations, and graphs of their solutions This will be an eploration of a variety of problems that occur when stuing rates of change. Many of these problems can be modeled

More information

Prelim 2 Math Please show your reasoning and all your work. This is a 90 minute exam. Calculators are not needed or permitted. Good luck!

Prelim 2 Math Please show your reasoning and all your work. This is a 90 minute exam. Calculators are not needed or permitted. Good luck! April 4, Prelim Math Please show your reasoning and all your work. This is a 9 minute exam. Calculators are not needed or permitted. Good luck! Trigonometric Formulas sin x sin x cos x cos (u + v) cos

More information

Ordinary Differential Equations

Ordinary Differential Equations Ordinary Differential Equations (MA102 Mathematics II) Shyamashree Upadhyay IIT Guwahati Shyamashree Upadhyay ( IIT Guwahati ) Ordinary Differential Equations 1 / 25 First order ODE s We will now discuss

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS HANDOUT DIFFERENTIAL EQUATIONS For International Class Nikenasih Binatari NIP. 19841019 200812 2 005 Mathematics Educational Department Faculty of Mathematics and Natural Sciences State University of Yogyakarta

More information

Section 11.1 What is a Differential Equation?

Section 11.1 What is a Differential Equation? 1 Section 11.1 What is a Differential Equation? Example 1 Suppose a ball is dropped from the top of a building of height 50 meters. Let h(t) denote the height of the ball after t seconds, then it is known

More information

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation.

Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: = 0 : homogeneous equation. Review For the Final: Problem 1 Find the general solutions of the following DEs. a) x 2 y xy y 2 = 0 solution: y y x y2 = 0 : homogeneous equation. x2 v = y dy, y = vx, and x v + x dv dx = v + v2. dx =

More information

Modeling with First-Order Equations

Modeling with First-Order Equations Modeling with First-Order Equations MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Radioactive Decay Radioactive decay takes place continuously. The number

More information

REFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3

REFERENCE: CROFT & DAVISON CHAPTER 20 BLOCKS 1-3 IV ORDINARY DIFFERENTIAL EQUATIONS REFERENCE: CROFT & DAVISON CHAPTER 0 BLOCKS 1-3 INTRODUCTION AND TERMINOLOGY INTRODUCTION A differential equation (d.e.) e) is an equation involving an unknown function

More information

Math 250 Skills Assessment Test

Math 250 Skills Assessment Test Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).

More information

Definition of differential equations and their classification. Methods of solution of first-order differential equations

Definition of differential equations and their classification. Methods of solution of first-order differential equations Introduction to differential equations: overview Definition of differential equations and their classification Solutions of differential equations Initial value problems Existence and uniqueness Mathematical

More information

Practice Exam 1 Solutions

Practice Exam 1 Solutions Practice Exam 1 Solutions 1a. Let S be the region bounded by y = x 3, y = 1, and x. Find the area of S. What is the volume of the solid obtained by rotating S about the line y = 1? Area A = Volume 1 1

More information

First order differential equations

First order differential equations First order differential equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. First

More information

2. (12 points) Find an equation for the line tangent to the graph of f(x) =

2. (12 points) Find an equation for the line tangent to the graph of f(x) = November 23, 2010 Name The total number of points available is 153 Throughout this test, show your work Throughout this test, you are expected to use calculus to solve problems Graphing calculator solutions

More information

FLAP M6.1 Introducing differential equations COPYRIGHT 1998 THE OPEN UNIVERSITY S570 V1.1

FLAP M6.1 Introducing differential equations COPYRIGHT 1998 THE OPEN UNIVERSITY S570 V1.1 F1 (a) This equation is of first order and of first degree. It is a linear equation, since the dependent variable y and its derivative only appear raised to the first power, and there are no products of

More information

Logarithmic Functions

Logarithmic Functions Metropolitan Community College The Natural Logarithmic Function The natural logarithmic function is defined on (0, ) as ln x = x 1 1 t dt. Example 1. Evaluate ln 1. Example 1. Evaluate ln 1. Solution.

More information

3.9 Derivatives of Exponential and Logarithmic Functions

3.9 Derivatives of Exponential and Logarithmic Functions 322 Chapter 3 Derivatives 3.9 Derivatives of Exponential and Logarithmic Functions Learning Objectives 3.9.1 Find the derivative of exponential functions. 3.9.2 Find the derivative of logarithmic functions.

More information

Elementary ODE Review

Elementary ODE Review Elementary ODE Review First Order ODEs First Order Equations Ordinary differential equations of the fm y F(x, y) () are called first der dinary differential equations. There are a variety of techniques

More information

Ch 4 Differentiation

Ch 4 Differentiation Ch 1 Partial fractions Ch 6 Integration Ch 2 Coordinate geometry C4 Ch 5 Vectors Ch 3 The binomial expansion Ch 4 Differentiation Chapter 1 Partial fractions We can add (or take away) two fractions only

More information

Math Spring 2014 Homework 2 solution

Math Spring 2014 Homework 2 solution Math 3-00 Spring 04 Homework solution.3/5 A tank initially contains 0 lb of salt in gal of weater. A salt solution flows into the tank at 3 gal/min and well-stirred out at the same rate. Inflow salt concentration

More information

The acceleration of gravity is constant (near the surface of the earth). So, for falling objects:

The acceleration of gravity is constant (near the surface of the earth). So, for falling objects: 1. Become familiar with a definition of and terminology involved with differential equations Calculus - Santowski. Solve differential equations with and without initial conditions 3. Apply differential

More information

x y

x y (a) The curve y = ax n, where a and n are constants, passes through the points (2.25, 27), (4, 64) and (6.25, p). Calculate the value of a, of n and of p. [5] (b) The mass, m grams, of a radioactive substance

More information

Consider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity

Consider an ideal pendulum as shown below. l θ is the angular acceleration θ is the angular velocity 1 Second Order Ordinary Differential Equations 1.1 The harmonic oscillator Consider an ideal pendulum as shown below. θ l Fr mg l θ is the angular acceleration θ is the angular velocity A point mass m

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Chapter Practice Test Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general solution to the eact differential equation. ) dy dt =

More information

Final Exam 2011 Winter Term 2 Solutions

Final Exam 2011 Winter Term 2 Solutions . (a Find the radius of convergence of the series: ( k k+ x k. Solution: Using the Ratio Test, we get: L = lim a k+ a k = lim ( k+ k+ x k+ ( k k+ x k = lim x = x. Note that the series converges for L

More information

Engg. Math. I. Unit-I. Differential Calculus

Engg. Math. I. Unit-I. Differential Calculus Dr. Satish Shukla 1 of 50 Engg. Math. I Unit-I Differential Calculus Syllabus: Limits of functions, continuous functions, uniform continuity, monotone and inverse functions. Differentiable functions, Rolle

More information

UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test

UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test UNIVERSITY OF HOUSTON HIGH SCHOOL MATHEMATICS CONTEST Spring 2018 Calculus Test NAME: SCHOOL: 1. Let f be some function for which you know only that if 0 < x < 1, then f(x) 5 < 0.1. Which of the following

More information

18.01 Calculus Jason Starr Fall 2005

18.01 Calculus Jason Starr Fall 2005 Lecture 17. October 1, 005 Homework. Problem Set 5 Part I: (a) and (b); Part II: Problem 1. Practice Problems. Course Reader: 3F 1, 3F, 3F 4, 3F 8. 1. Ordinary differential equations. An ordinary differential

More information

AP Calculus Testbank (Chapter 6) (Mr. Surowski)

AP Calculus Testbank (Chapter 6) (Mr. Surowski) AP Calculus Testbank (Chapter 6) (Mr. Surowski) Part I. Multiple-Choice Questions 1. Suppose that f is an odd differentiable function. Then (A) f(1); (B) f (1) (C) f(1) f( 1) (D) 0 (E). 1 1 xf (x) =. The

More information

MATH 1231 MATHEMATICS 1B Calculus Section 3A: - First order ODEs.

MATH 1231 MATHEMATICS 1B Calculus Section 3A: - First order ODEs. MATH 1231 MATHEMATICS 1B 2010. For use in Dr Chris Tisdell s lectures. Calculus Section 3A: - First order ODEs. Created and compiled by Chris Tisdell S1: What is an ODE? S2: Motivation S3: Types and orders

More information

MATH 391 Test 1 Fall, (1) (12 points each)compute the general solution of each of the following differential equations: = 4x 2y.

MATH 391 Test 1 Fall, (1) (12 points each)compute the general solution of each of the following differential equations: = 4x 2y. MATH 391 Test 1 Fall, 2018 (1) (12 points each)compute the general solution of each of the following differential equations: (a) (b) x dy dx + xy = x2 + y. (x + y) dy dx = 4x 2y. (c) yy + (y ) 2 = 0 (y

More information

MATH 312 Section 2.4: Exact Differential Equations

MATH 312 Section 2.4: Exact Differential Equations MATH 312 Section 2.4: Exact Differential Equations Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline 1 Exact Differential Equations 2 Solving an Exact DE 3 Making a DE Exact 4 Conclusion

More information

Ma 221 Final Exam Solutions 5/14/13

Ma 221 Final Exam Solutions 5/14/13 Ma 221 Final Exam Solutions 5/14/13 1. Solve (a) (8 pts) Solution: The equation is separable. dy dx exy y 1 y0 0 y 1e y dy e x dx y 1e y dy e x dx ye y e y dy e x dx ye y e y e y e x c The last step comes

More information

Fourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π

Fourier transforms. c n e inπx. f (x) = Write same thing in an equivalent form, using n = 1, f (x) = l π Fourier transforms We can imagine our periodic function having periodicity taken to the limits ± In this case, the function f (x) is not necessarily periodic, but we can still use Fourier transforms (related

More information

First Midterm Examination

First Midterm Examination Çankaya University Department of Mathematics 016-017 Fall Semester MATH 155 - Calculus for Engineering I First Midterm Eamination 1) Find the domain and range of the following functions. Eplain your solution.

More information

JUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) A.J.Hobson

JUST THE MATHS UNIT NUMBER ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) A.J.Hobson JUST THE MATHS UNIT NUMBER 15.3 ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) by A.J.Hobson 15.3.1 Linear equations 15.3.2 Bernouilli s equation 15.3.3 Exercises 15.3.4 Answers to exercises

More information

2.2 Separable Equations

2.2 Separable Equations 2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve

More information

Solution: APPM 1350 Final Exam Spring 2014

Solution: APPM 1350 Final Exam Spring 2014 APPM 135 Final Exam Spring 214 1. (a) (5 pts. each) Find the following derivatives, f (x), for the f given: (a) f(x) = x 2 sin 1 (x 2 ) (b) f(x) = 1 1 + x 2 (c) f(x) = x ln x (d) f(x) = x x d (b) (15 pts)

More information

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx.

Mathematics 104 Fall Term 2006 Solutions to Final Exam. sin(ln t) dt = e x sin(x) dx. Mathematics 14 Fall Term 26 Solutions to Final Exam 1. Evaluate sin(ln t) dt. Solution. We first make the substitution t = e x, for which dt = e x. This gives sin(ln t) dt = e x sin(x). To evaluate the

More information

MATH 31BH Homework 5 Solutions

MATH 31BH Homework 5 Solutions MATH 3BH Homework 5 Solutions February 4, 204 Problem.8.2 (a) Let x t f y = x 2 + y 2 + 2z 2 and g(t) = t 2. z t 3 Then by the chain rule a a a D(g f) b = Dg f b Df b c c c = [Dg(a 2 + b 2 + 2c 2 )] [

More information

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that

More information

DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =

More information