LECTURE 4-1 INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS

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1 130 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 equation is ordinary if the unknown function depends on only one variable; x 2 d2 y dy + 3x dx2 dx + y = 4 A differential equation is a partial Differential equation if it depends on more than one variable; u x = 2 u x 2 INDEPENDENT/DEPENDENT VARIABLES: The variable with respect to which we are differentiating is the independent variable, the other is the dependent variable. In addition to the dependent/independent variables, a third type of variable, called a parameter, may appear in the equation. dy dx = 3x y + h x independent variable, y dependent variable, h: parameter ORDER OF DIFFERENTIAL EQUATION: The order of a differential equation is the order of the highest order derivative appearing in the differential equation. y + 2y = x ; 2 nd order x 4 y x = 0 ; 1 st order MEANING OF SOLUTION: An analytic solution of a differential equation is a sufficiently differentiable function that, if substituted into the equation, together with the necessary derivatives, make the equation an identity over some interval of the independent variable. If we can write a solution y = φ(x), this solution is called explicit solution. If not, it is called implicit solution. EXAMPLE 1 EXAMPLE 2 What is the anti-derivative of f(x) = 2x? Is there at least one solution of a differential equation y = f(x, y)? y = x : it has at least one solution y + y 2 = x : it does not have a solution

2 131 EXAMPLE 3 EXAMPLE 4 Verify that y = e 2x is a solution of y + 3y + 2y = 0 Show that y(x) = 1 x 2 is a solution of y = x on y on the interval (, ). the interval (1, ). LINEAR/NON-LINEAR DE: A differential equation is linear if the unknown function and its derivative appear linearly, otherwise it is non-linear. y + 3y = 2; linear x 2 y + y = 0; linear y + yy = 0; non-linear (y ) 2 + 3y = 1; non-linear AUTONOMOUS DE: An autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. y + 4y + 5 = 0: autonomous, y + y = x: Not-autonomous HOMOGENEOUS DE: Homogeneous differential equations involve only derivatives of y and terms involving y, and they're set to 0, as in this equation: y + 3y + 3xy = 0 : homogeneous, y + 3y + 3xy = x + 2 : Non-homogeneous EXAMPLE 5 Determine the order of the differential equations and whether it is autonomous, linear, or homogeneous. (A) dy + 5y = ex dx (B) (1 + y)y + 2y = 4x (C) 2y + y 2 = 0 (D) xy (sin x)y + e x y = 0

3 132 DEFINITION: Graphical/Numerical technique 1) A SLOPE FIELD: a slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation 2) INTEGRAL CURVE: A solution curve drawn in the direction field whose tangent at every point is parallel to the line segment at that point is called an integral curve (or trajectories), and it is a particular solution to the DE. All solutions have this tangency property! 3) ISOCLINES: Consider the DE dy = f(x, y). If f is simple enough, we may find curve (isoclines) in the xy- dx plane along which dy is constant. Doing so, is the Method of isoclines. dx 4) EQUILIBRIUM SOLUTIONS: An equilibrium solution (also called steady-state solution) is the value of that does not change over time: y = 0 o o o An equilibrium is unstable if solutions near the equilibrium eventually move away from that equilibrium. An equilibrium is stable if solutions near the equilibrium move closer to that equilibrium. Equilibrium solutions in which solutions that start near them move toward the equilibrium solution are called asymptotically stable equilibrium points or asymptotically stable equilibrium solutions Analytic View y = f(x, y) Direction field Geometric View y 1 (x) is a solution at a point (x 0, y 0 ) Integral curve at a point (x 0, y 0 ) EXAMPLE 6 Draw the direction field of dy = y x dx

4 133 EXAMPLE 7 EXAMPLE 8 Sketch the direction field of dy dx = y + x Sketch the direction field of y = y 2.

5 134 EULER S METHOD: When y = f(x, y) If our step size h = x n+1 x n is fixed, then x n+1 = x n + h (n = 0,1,2, ) y 0 = f(t 0 ) y actual value error approximate vale y n+1 = y n + f(x n, y n )h tangent line at x=x n x n x n+1 step size h =x n+1 x n x EXAMPLE 9 EXAMPLE 10 Consider the ODE y = y x and y(1) = 1. Can we Use Euler s method to approximate solution of the IVP approximate y(1.5) with the step-size of h = 0.1? with x = 0, 0.1, 0.2, 0.3, 0.4 y = 3 + x y, y(0) = 1

6 135 LECTURE 4-2 SEPARABLE METHOD DEFINITION: A first order differential equation of the form dy = g(x) h(y) dx is said to be separable or to have separable variables. SEPARABLE dy dx = y2 e x NON-SEPARABLE dy = y + sin(x) dx By dividing by the function h(y), 1 h(y) dy dx = g(x) By letting p(y) = 1 h(y), p(y) dy dx = g(x) If y = φ(x) represent a solution of the above equation, p(φ(x))φ (x) = g(x) p(φ(x)) φ (x)dx =dy p(y) dy = g(x) dx = g(x) dx EXAMPLE 1 EXAMPLE 2 Solve the differential equation: (1 + x)dy ydx = 0 Find the explicit solution of the initial value problem; dy dx = x y, y(4) = 3

7 136 EXAMPLE 3 EXAMPLE 4 Solve the differential equation: (A) dy dx + 2xy2 = 0 Solve the differential equation: (A) yy = (x + xy 2 )e x2 (B) xy + 2y = 0 (B) dy = (y 2 1)dx

8 137 EXAMPLE 5 EXAMPLE 6 Find the explicit solution of the initial value problem; dy = xy, y(0) = 3 dx Find the explicit solution of the initial value problem; dy = y 2xy, dx y(1) = 1

9 138 LECTURE 4-3 INTEGRATING FACTOR METHOD LINEAR EQUATION: General Form Standard Form : a(x)y + b(x)y = c(x) : y + p(x)y = g(x) KEY IDEA: When confronted with the 1 st order linear DE y + p(x)y = g(x), a < x < b Try to find a function μ(x) defined on a < x < b (with μ(x) 0 for a < x < b) so that μ(x)[y + p(x)y] (μ(x)y) = μ(x)g(x) This requires LHS = (μ(x)y) = μ(x)y + μ(x) y μ (x) = p(x)μ(x) Then μ (x) μ(x) = p(x) ln μ(x) = p(x) dx y μ(x) = μ(x)g(x) dx y = This μ(x) = exp( p(x) dx) is called the integrating factor. μ(x) = exp ( p(x) dx) 1 μ(x)g(x) dx μ(x) EXAMPLE 1 Solve the differential equation; xy 2y = x 2

10 139 EXAMPLE 2 EXAMPLE 3 Solve the differential equation; Solve the differential equation; (A) y + y = e x xy y = x 3 e x (B) y + 2xy = 10x

11 140 EXAMPLE 4 EXAMPLE 5 Solve the differential equation; Solve the differential equation; y 2y = x 3 e 2x xy + y = cos(x)

12 141 EXAMPLE 6 EXAMPLE 7 Solve the differential equation; Solve the differential equation; y = y + e 2x y(0) = 1 xy + y = 3x 2 x, y(1) = 0

13 142 LECTURE 4-4 APPLICATIONS NATURAL GROWTH AND DECAY: The differential equation dy = ky (k a constant) dx y = Cekx Serves as a mathematical model for a remarkably wide range of natural phenomena: any involving a quantity whose time rate of change is proportional to its current size. Population growth Compound interest Radioactive decay Drug elimination EXAMPLE 1 EXAMPLE 2 The plutonium has 24,000 years as half-life. Ten gram of the plutonium were released in an accident. How long will it take for the 10 grams to decay to 1 gram? An experiment population of fruit flies increasing according to the law of exponential growth. There were 100 flies after the second day of the experiment and 200 flies after the fourth day. How many flies were in the original population?

14 143 NEWTON S LAW OF COOLING/WARMING: The rate at which the temperature of a body changes is proportional to the difference between the temperature of the body and the temperature of the surrounding medium. dt dt = k(t T m) T = T m + Ce kt T(t) represents the temperature of a body at time t, T m the temperature of the surrounding medium, and k is a constant of proportionality. EXAMPLE 3 EXAMPLE 4 A pitcher of buttermilk initially at 25 is to be cooled by My coffee is 120 when class starts, and the classroom is setting it on the front porch, where the temperature is After 30 minutes, the coffee is 100. Suppose that the temperature of the buttermilk has (A) What will the temperature of the coffee be at the end dropped to 15 after 20 minutes. When will it be at 5? of class (50 min)? (B) Suppose it was brewed at 160. When did I brew it?

15 144 NOTE: Conservation principles often lead to equations of the form By conservation, I mean the following picture; dq dt = rate in rate out rate in Q(t) box rate out where Q(t) denotes the amount of substance (i.e., chemicals, people, money, etc ) in the box at time t, (Chemical mixture problems) In mixture problems the relevant quantities are Volume flow rate, units = volume/time. Concentration, units = mass/volume Mass flow rate, units = mass/time When using a consistent system of units, for example volume in liters, mass in kg, time in minutes, then (Volume flow rate)(concentration) = mass flow rate ; ( L min ) (kg L ) = kg = mass flow rate min EXAMPLE 5 A tank initially contains 150 gal of fresh water. Brine containing 2 lb of salt per gallon enters the tank at the rate of 3 gal/min, and the well mixed brine in the tank flows at the rate of 3 gal/min. What is the amount of salt in the tank at time t?

16 145 EXAMPLE 6 EXAMPLE 7 A 300 gal-tank initially contains 150 gal of fresh water. A 400-gal tank initially contains 100 gal of brine Brine containing 2 lb of salt per gallon enters the tank at containing 50 lb of salt. Brine containing 1 lb of salt per the rate of 3 gal/min, and the well mixed brine in the tank gallon enters the tank at the rate of 5 gal/s, and the well flows at the rate of 1 gal/min. What is the amount of salt mixed brine in the tank flows at the rate of 3 gal/s. How in the tank at time t? much salt will the tank contain when it is full of brine?

17 146 LECTURE 4-5 SEQUENCE SEQUENCE: 1) a 1, a 2, a 3, is an infinite sequence where each a i is real number, denoted by {a n } n=1 or {a n }. In other words, an infinite sequence is a function whose domain is the set of positive integers. 2) The element a m of {a n } is m-th term of the sequence. EXAMPLE 1 EXAMPLE 2 Find the four terms of the following sequence: Find a formula for the general term of the following (A) {a n } = {3 + ( 1) n } sequence (A) 1, 4, 7, 10, 13, (B) {a n } = { n+3 2n } (B) 1, 2 3, 4 9, 8 27, (C) 3, 1, 3, 1, 3, 1, (C) {a n } = {( 1) n n } n+1 (D) 3 5, 4 25, 5 125, 6 625,

18 147 PROPERTIES OF LIMIT: Suppose that c is a constant and lim a n = L and lim b n = M exists. 1) lim (a n ± b n ) = L ± M 2) lim c a n = cl 3) lim (a n b n ) = LM 4) lim ( a n ) = L, M 0 b n M 5) lim (a n ) p = L p if p > 0 and a n > 0 6) lim r n = { 0, r < 1 1, r = 1 does not exist, otherwise CONVERGENCE: A sequence {a n } has the limit L and we write lim a n = L make the terms an as close to L as we like by taking n sufficient large. or a n L as n if we can 1) If lim a n exists, we say that {a n } is convergent. Otherwise, we say that {a n } is divergent. 2) If lim a n exists, {a n } is absolutely convergent. 3) If {a n } is convergent but not absolutely convergent, it is called conditionally convergent. 4) Warning: An absolute convergence sequence is not always a convergence sequence. How to find the limit of sequence {a n }? (A) Use the graph of the sequence (B) Geometric series {a n = r n } case: lim r n = { (C) Use lim 1 n p = 0 (p > 0) and properties of limit 0, r < 1 1, r = 1 does not exist, otherwise (D) (L Hopital Rule) Let f and g be differentiable functions such that a n = f(n) g(n) 0 0 f(n) lim g(n) = lim f (n) g (n) (E) (sandwich Theorem) Let {a n }, {b n }, and {c n } be sequence such that ± (or ) as n, then ± b n a n c n lim b n = L and lim c n = L exists. Then lim a n = L

19 148 EXAMPLE 3 EXAMPLE 4 Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. (A) a n = 4n 1 3n+2 Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. (A) a n = n2 3 n (B) a n = 2n2 4 4n+2 (B) a n = ln(n 3 + 1) ln(3n n) (C) a n = 3n2 +2n 5n 3 4n+2 (C) a n = 1 ( 1) n (D) a n = (n 2)(n+3) (2n 1)(n+1) (D) a n = cos(2nπ)

20 149 EXAMPLE 5 EXAMPLE 6 Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. (A) a n = ( 2 5 )n Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. (A) a n = en 5n (B) a n = ( 4 3 )n (C) a n = ( 3 4 )n 2 (B) a n = e n n 2 (D) a n = 2n 5 n +4 n (E) a n = 3n +5 4 n 2 2n 3 n

21 150 EXAMPLE 7 EXAMPLE 8 Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. (A) a n = e n cos(n) Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. a n = ( 5 + n n n ) (B) a n = sin(nπ 2 ) n+2

22 151 EXAMPLE 9 EXAMPLE 10 Given the general term of the sequence, decide whether the sequence is convergent or divergent. If it is convergent, find the limit. a n = (1 4 n ) n Decide whether the following sequence is convergent or divergent. (A) 1 2, 2 3, 3 4, 4 5,, n n+1, (B) 1, 4, 7, 10,, 3n 2, (C) 3, 3, 3, 3, 3,, 3, (D) 1, 2, 3, 4,, ( 1) n n,

23 152 EXAMPLE 11 Decide whether the following sequence absolutely converges, converges or diverges (A) { ( 1)n n } (B) {( 3) n } (C) {( 1 2 )n } (D) {sin (nπ + π 2 )} (E) {(1 1 2 ) + ( ) + ( ) + ( 1 n 1 1 n )} (F) { ( 1)n n 2 5n 3 } MONOTONIC SEQUENCE: A sequence {a n } is monotonic when its terms are non-decreasing or non-increasing; a n a n+1 (or a n a n+1 ) for all n = 1,2,3, BOUNDED SEQUENCE: 1) A sequence {a n } is bounded above when there is a real number M such that a n < M for all n. 2) A sequence {a n } is bounded below when there is a real number M such that a n > M for all n. 3) A sequence {a n } is bounded when there is a real number M such that a n < M for all n. THEOREM (BOUNDED MONOTONIC SEQUENCES): If A sequence {a n } is bounded and monotonic, then it is convergent.

24 153 EXAMPLE 12 Given the general term of the sequence, decide whether the sequence is monotonic and bounded. (A) a n = ( 1 3 )n (B) a n = 5 + ( 1) n (C) a n = n 2n 1 (D) a n = 2n + 1 (E) a n = ( 2 3 )n (F) a n = ( 4) n

25 154 PRACTICE PROBLEMS FOR UNIT 4 1. Decide the order of the following differential equation: (A) y + 4y y = x 3 + e 2x (B) y y = x 2 sin(3x) (C) (y ) 3 y = 5 2. Verify that y(x) = 2e x x 2 is a solution to the IVP: y y = x 2 2x, y(0) = Plot a direction field for the differential equation y = x y near the point (1,1). 4. Solve the ordinary differential equation or IVP: (A) e y x 2 y 3 = 0 (C) xy 2y = 4x 2 (B) dy dx = y x2 y, y(1) = 1 (D) x 3 y + 4x 2 y = e x (E) y + 5y = x 2 + e x, y(0) = 1 (F) y + y = y 2 (G) dy dx + 3x2 y = 6x 2 (H) x 2 y + xy = 1, x > 0, y(1) = 2 5. Suppose that $1200 is invested at a rate of 5%, compounded continuously. (A) Assuming no additional withdrawals or deposits, how much will be in the account after 10 years? (B) How long will it take the balance to reach $5000? 6. The tank hold a well-mixed 100-gallon saline solution with a total of 10 pounds of salt. Fresh water comes in at a rate of 2 gallons/minute, and the well-mixed solution leaves at a rate 2 gallons/minute. Find the amount of salf in the tank at time t. 7. In a certain chemical process, a tank whose capacity is 500 gallons contains 50 pounds of a chemical dissolved in 100 gallons of water. Now, chemical solution containing 1 pound/gallon of the chemical, is pumped into the tank at a rate of 3 gallons/minute. The well-mixed mixture is pumped out of the tank at a rate of 2 gallons/minute. Find a formula for chemical in pounds at time t in the tank 8. Suppose that an object initially having a temperature of 20 is placed in a large temperature controlled room of 80 and one hour later the object has a temperature of 35. What will its temperature be after three hours? 9. (Euler s Method) Use Euler s Method to fill (A)~(E) in the following table for the IVP: y = f(x, y) = x + y 2, y(0) = 1 n x y f(x, y) (A) (B) (C) (D) (E)

26 Write the first three terms of the sequence: a n = 1 3 n + 5 n Write an expression for the n-th term of the sequence: 10, 22, 34, 46, 58,. 12. Determine the convergence or divergence of the sequence with the given n-th term. Find the limit of each convergent sequence. (A) a n = 3n 4 n (B) a n = 4 n2 10n 9 (C) a n = n! n n

27 156 SOLUTIONS 1. (A) second order (B) first order (C) first order 2. y(0) = 2e = 2 and y y = (2e x x 2 ) (2e x x 2 ) = 2e x 2x 2e x + x 2 = x 2 2x. Therefore, it is a solution (A) e y = x 3 + C (B) y = e (x x ) (C) y = 4x 2 ln x + Cx 2 (D) y = (x+1) + C (E) y = 1 6 ex x x e 5x (F) y = 1 1 De x 5. (A) 1200e (B) ln( ) 6. m(0) = 10, m = m m = 10e t/ years 7. m(0) = 50, m = 3 m m = (100 + t) 100+t t 8. T = 80 60e t ln(3 4 ) T(3) = (A)1.05 (B)1.15 (C)1.22 (D)1.70 (E)1.39 x 4 e x 10. a 1 = 3; a 2 = 3 4 ; a 3 = a n = 12n (A) lim a n = lim ( 3 4 )n = 0 since the common ratio 3 of the geometric sequence is less than 1. 4 (B)Divergence (C)Since 0 n! n(n 1) 2 1 nn = 1 and by comparison test, lim n n n n n 1 n = 0. So it converges to 0.