**Step-1**

Take Laplace transform on both the sides of the given differential equation.

**Step-2**

The necessary formulas of Laplace transform of the derivatives of a function are applied.

Most commonly used are:

**Step-3**

Substituting all the necessary Laplace transform, the expression of is obtained.

**Step-4**

Once is found, then can be obtained by taking Inverse Laplace transform.

## Topic

Steps to find the solution of ordinary differential equations using Laplace Transforms

## Examples

- Solve the differential equation using Laplace transform (d^2 x)/(dt^2 )+3 dx/dt+2x=e^(-t) given the initial conditions x(0)=0 & x'(0)=1
- Solve the equations (d^2 y)/ (dt^2) +5 dy/dt+6y=5e^2t given that y (0) =2, dy/dt (0) =1
- Solve the equation (d^2 y)/ (dt^2) +2 dy/dt-3y=sin t under the conditions y(0) = dy/dt (0) =0
- Solve the equation (d^2 y)/ (dt^2) – dy/dt = 0 under the conditions y(0) = y’(0) =3
- Solve the equation (d^2 y)/ (dt^2) + 3 dy/dt + 2 y(t) = 0 under the conditions y(0) = 1 & y’(0) =0
- Solve the initial problem y”+2y’+2y = 5 sin t when y(0)=y'(0)=0