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 and its derivative. Ordinary Differential Equations Page 1
Example 1 In the d.e. = 7 x + 4x+ 3 y f (x) is the unknown function. is the derivative of the unknown function. Ordinary Differential Equations Page
The d.e. is ordinary (i.e. an o.d.e.) if the unknown function depends on only one independent variable. Example The followings are o.d.e. (a) (b) (c) (d) = = dt e y d y ( ) x 3-4 x+ 1 t + ( ) + 3( ) 3 6 = 1 + y = 5x Ordinary Differential Equations Page 3
Order and Degree The order of a d.e. is the order of the highest derivative of the d.e. The degree of a d.e. is the highest power of the highest derivative. Example 3 Refer to the d.e. in Example, find the order and degree of the d.e. Ordinary Differential Equations Page 4
LINEAR DIFFERENTIAL EQUATIONS A d.e. is said to be linear if the dependent variable and its derivative are of degree one and there are no product involving the dependent variable and the derivative. Example 4 Determine whether the following d.e. are linear. (a) + xy = 0 (b) y + 3xy - = 0 (c) d y + 3x - = 0 (d) ( ) + xy = 0 Ordinary Differential Equations Page 5
Example 5 Form a d.e. of the problem given that velocity of a bo is proportional to its displacement. Example 6 Form a d.e. of the problem that a resistor R and a capacitor C is connected in series with an a.c. source of 50 sin ( f t). Ordinary Differential Equations Page 6
Solution of a Differential Equation A solution of a d.e. which involves the independent variable x and dependent variable y is the finding of the equation f(x, y) = 0 which satisfies the d.e. Example 7 Determine whether the equation (a) (b) y= e -x + xe -x -x y= 3e - 4 xe -x and d y are solutions of the d.e. y= 0. Ordinary Differential Equations Page 7
A general solution of a d.e. is a set of all possible solutions of the d.e. A particular solution of a d.e. is any one solution of the d.e that fulfills the specific boundary conditions. Example 8 Given the d.e. = 10 x y = 5x + C is a general solution while y=5x + 3 is a particular solution. Ordinary Differential Equations Page 8
Initial Conditions and Boundary Conditions These two conditions determine the value(s) of the constant(s) in the general solution and form the particular solution. Example 9 Determine the values of the constants C 1 and C in y = C 1 sin x + C cos x with the following conditions. (a) The initial conditions y(0) = 1, y (0) =. (b) The boundary conditions y(0) = 1 and y ( ) =. Ordinary Differential Equations Page 9
Direct Integration As integration is the reverse process of differentiation a d.e. may be solved by direct integration. Example 10 Find the general solution of the d.e. (a) (b) = 4 x y = e d - x Ordinary Differential Equations Page 10
Example 11 Find the particular solution of the d.e. = - sin x which satisfies the condition that y = when x = 0. Ordinary Differential Equations Page 11
FIRST ORDER ORDINARY DIFFERENTIAL EQUATION Method of Separation of Variables Definition A de d.e. is called variables separable if it may be written in the form or =f ( x ) g ( y ) f ( x ) = g( y) and the general solution is 1 g ( y ) = f ( x ) + C Ordinary Differential Equations Page 1
Example 1 Determine whether the following d.e. are variable separable. (a) (b) = = x ( x + )( y - 3) y + y - 6 (c) = 3 x - 3y+ 5 4 (d) = x y - 3x y - 3 Ordinary Differential Equations Page 13
Example 13 Find the general solution of () (a) sin 5x = y (b) -5 x = y e (c) - e 5 x = (d) () x y' = y 0 Ordinary Differential Equations Page 14
Example 14 Solve the d.e. in Example 13(b) and (d) given (a) in Example 13(b) y(0) = e. (b) in Example 13(d) y (4) = 1. Ordinary Differential Equations Page 15
First Order Linear ODE The standard form of a first order linear d.e. may be written in + P( x) y =Q( x) Example 15 What are P and Q referred to the standard form? (a) (b) t - 3x + dt y - 3t 3 5x = 0 x + t = 5x Ordinary Differential Equations Page 16
Integrating Factor The linear d.e. in standard form + P( x) y =Q( x) has an integrating factor μ(x) = e P(x) The general solution of the linear d.e. is μ ( x) y = μ( x) Q( x) + C Ordinary Differential Equations Page 17
Example 16 Find the general solution of the following linear d.e. (a) (b) (c) (d) = 5y x + y = x + y tan x = 4 x x -3y = x e given sin x given y y 6 when when x x 6 1 Ordinary Differential Equations Page 18
First Order O.D.E. Applications Example 17 The resistance force on a bo is directly proportional p to the velocity of the bo. Set up a differential equation and find the velocity of a bo of mass.4 kg after 10 s given its initial velocity 30 ms -1 and constant of proportionality 0.7 kgs -1. Ordinary Differential Equations Page 19
Example 18 A charged capacitor C is discharged through a resistor R. Set up a differential equation for discharging the capacitor. A capacitor of 0.5 F is first charged to a potential of 70 V and is then discharged through a resistor of 0k ohms. Find the potential across the capacitor after 10 ms. Ordinary Differential Equations Page 0
SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS Reference: Croft & Davison Chapter 0 Blocks 5 6 Terminology A second order linear d.e. has the general form b d y ( x) +b1( x) +b 0 ( x) y = f ( x). If f ( x) 0 the d.e. is homogeneous. If f ( x) 0 the d.e. is inhomogeneous. Ordinary Differential Equations Page 1
If b i (x) are constants, where i = 0, 1,, the d.e. is said to be with constant coefficients. A Second Order Linear D.E with Constant Coefficients has the form where p, q are constants d y + p + q y = f ( x) Ordinary Differential Equations Page
Example 19 Which of the following are constant coefficient d.e.? Which are homogeneous? (a) d y (b) d y (c) d y - 3 + 5x - 3 = 5y - = 6x 4y = 0 Ordinary Differential Equations Page 3
General Solution In order to solve the d.e. d y +p +qy=f ( x ) it is necessary to first solve the homogeneous de d.e. d y + p + qy = 0 The solution the homogeneous d.e. is called the complementary function, y cf and it contains two arbitrary constants. Ordinary Differential Equations Page 4
Any solution satisfied the inhomogenous d.e. is called particular integral, y pi. The general solution of the inhomogenous d.e. is the sum of the complementary function and the particular integral. y=y y cf + y pi Ordinary Differential Equations Page 5
Example 0 y Asin x B cos x Show that is a complementary function of 4 y= 0. Note: If 1 (x) and (x) are two linear dependent solutions of the homogeneous d.e. then the linear combination C 1 1 (x) + C (x) is also a general solution of the d.e. Ordinary Differential Equations Page 6
Example 1 Given the d.e. d y + 7 + 1 y = 0. Show that (a) -3x e is a solution (b) e - 4 x is a solution (c) C e 1-3x + C e -4 x is a complement ary solution Ordinary Differential Equations Page 7
The d.e. d y + p + qy = 0 can be written by using the D operator, where d D denotes and D denotes d. Rewrite the equation in the form ( D + pd + q) y = 0 Ordinary Differential Equations Page 8
Solution of Homogenous Second Order Linear D.E with Constant Coefficients Complementary Function The general solution of the d.e. d y + p + qy = 0 is y C1 1(x) C (x) k1x kx where (x) 1 e and (x) e are solutions of the differential equation provided that ( ) and are linearly independent. ( x ) 1 x Ordinary Differential Equations Page 9
Let kx y e be the general solution of the d.e., then kx d y ke and k e kx Hence the d.e. becomes e kx ( k pk q) 0 e kx 0 Since, therefore we have k pk q 0 - an auxiliary equation of the d.e. and is quadratic in k. Ordinary Differential Equations Page 30
Example Write down the auxiliary equation of the d.e. and find the roots of the auxiliary equation. (a) () d y (b) d y (c) d y - + + 7 8 + + + 1 y = 0. 16 y = 0. y= 0. Ordinary Differential Equations Page 31
Nature of Roots of the Auxiliary Equation (1) Roots are real and distinct If the roots of the auxiliary equation are and k, then the general solution of the d.e. is k1x kx y = C 1 e + C e k 1 Example 3 Solve the d.e. d y (a) - d y (b) - + 8y= 0. - 6 y = 0. Ordinary Differential Equations Page 3
() Roots are equal If the repeated roots of the auxiliary equation are k,then the general solution of the d.e. is or y = y = C 1 e kx + C ( C1 + C x) e xe kx kx Example 4 d y Solve the d.e. (a) + 8 + 16 y = 0. d y (b) - 14 + 49 y = 0 Ordinary Differential Equations Page 33
3. Roots are complex The solution is (k1+k j)x (k1-k j)x y = A e + B e However, it is usually written as y = e k x ( C1 cos k x + C sin k 1 x where A, B, C 1 and C are constants to be determined by boundary conditions. ) Example 5 Solve the d.e. () (a) D y + Dy + y = 0. (b) D y + 5 y = 0. (c) D y + 6 Dy + 13 y = 0 given when x= 0, y= and y' = 1. Ordinary Differential Equations Page 34
Particular Integral of inhomogenous ODE with constant coefficients i For a inhomogeneous second order ODE, d y + p + qy = f( x) Particular integral is any function that when substituted into the left-hand side and simplified, results in the function on the right. Ordinary Differential Equations Page 35
Inhomogenous term f(x) constant Trial solution constant ax r + +bx+c C 1 x r + +C r x+c r+1 cos kx sin kx e kx cosh kx C 1 cos kx + C sin kx C 1 cos kx +Csin kx C 1 e kx C 1 cosh kx + C sinh kx sinh hkx C 1 cosh kx +C sinh hkx Ordinary Differential Equations Page 36
Example 6 Find the particular integral of d y x - - 6y = e x Let y( x) = C x d y x ; then 1 e C e C e 1, and 4 1 x x x 4 C1 e C1 e 6 C1 e Hence 4C e 1 x e x x e C 1 1 4 Therefore y The general solution is y pi 1 e x 4 Ae Be 3x x 1 4 e x Ordinary Differential Equations Page 37