Solutions of Differential Equations by Laplace Transform

Solutions of Differential Equations by Laplace Transform

The method of Laplace transform is very useful for solving linear differential equations with constant coefficients and

associated initial conditions.

To perform this task, we take the Laplace transform of the given differential equation, making use of the initial

conditions. This lead to an algebraic equation in the Laplace transform of the required solution. After rearrange the

Laplace transform of the solution and taking the inverse Laplace transform, the required solution is obtained.


Example 29: Solve the following differential equation using Laplace transform:

y^ +2y=10e^3t

with y(0)=6.

Solution:

Taking Laplace transform of the differential equation, we get:

[sY(s)-y(0)]+2Y(s)=10/(s-3)

With y(0)=6, then:

sY(s)-6+2Y(s)=10/(s-3)

?? Y(s)(s+2)=10/(s-3)+6

?? Y(s)(s+2)=(10+6(s-3))/(s-3)

?? Y(s)=(10+6(s-3))/(s+2)(s-3) =(10+6s-18)/(s+2)(s-3) =(6s-8)/(s+2)(s-3)
To perform this task, we take the Laplace transform of the given differential equation, making use of the initial conditions. This lead to an algebraic equation in the Laplace transform of the required solution. After rearrange the Laplace transform of the solution and taking the inverse Laplace transform, the required solution is obtained.