1550078481-Ordinary_Differential_Equations__Roberts_

246 Ordinary Differential Equations

Apply the Inverse Laplace Transform

Applying the inverse Laplace transform to (7) and using the linearity prop-

erty of the inverse Laplace transform, yields

(8) y(x)=c1L -1 [ --1 ] +c2L -1 [ --1 ] --£^1 -1 [ 1] -

s+2 s-l 4 s

-~£-1 [2-J -£-1 [2-J

2 s^2 s^3

-2x x 1 1 1 2

= c 1 e + c2e -

4

• 2 x - 2 x ·

Hence, the general solution of the differential equation y" + y' - 2y = x^2 - 1

is

y x ( ) = c 1 e -2x + c2e x - 4 1 - 2x^1 -^1 2x^2

where c 1 and c 2 are arbitrary real constants.

The following example illustrates how to use the Laplace transform method
to solve an initial value problem.

EXAMPLE 3 Using the Laplace Transform to Solve an

Initial Value Problem

Find the solution of the initial value problem

y" - 4y' + 5y = 2ex - sinx; y(O) = 1, y'(O) = -1

using the Laplace transform method.

SOLUTION

Apply the Laplace Transform to the Differential Equation

The Laplace transform of the given differential equation is

(9)

(-y' (O)-sy(O)+s^2 L[y(x)])-4(-y(O)+sL[y(x)]) +5L[y(x)] = -

2

• 
  
  
  
  • 
    
    
    
    • 2 -
    
    
    
    
  
  
  
  

1

-.
s- 1 s +1
Solve the Algebraic Equation for L[y(x)]
Substituting the given initial conditions into (9) and solving the resulting
equation for L[y(x )], we find

2 1

s-5+-----

L[y(x) ] = s - 1 s^2 + 1

s^2 - 4s + 5

s^4 - 6s^3 + 8s^2 - 7 s + 8

(s - l)(s^2 + l)(s^2 - 4s + 5) ·