Step 1:

The differential equation is $\"\"$ .

Solving non-homogenous differential equation:

If the differential equation is in the form of $\"\"$ , then general solution of the non-homogenous differential equation is $\"\"$ , where $\"\"$ is the general solution of the complementary equation and $\"\"$ is the particular solution.

General solution of the complementary equation:

If the differential equation is in the form of $\"\"$ , then general solution of the complementary equation is $\"\"$

Particular solution of the differential equation :

If the differential equation is in the form of $\"\"$ then the particular solution of the equation is $\"\"$ , where

$\"\"$ and $\"\"$ .

Here $\"\"$ is the wronskian of $\"\"$ and $\"\"$ .

$\"\"$ .

Step 2:

Consider the auxiliary equation of the differential equation.

$\"\"$

Since the root of the equations are real and distinct then the solution of the differential equation is $\"\"$ .

The particular solution of the differential equation is in the form of $\"\"$ , where $\"\"$ and $\"\"$ .

Find wronskian of $\"\"$ and $\"\"$ is

$\"\"$

Step 3:

Find $\"\"$ .

$\"\"$

Re-write the expression.

$\"\"$

Find $\"\"$ .

$\"\"$

Re-write the expression.

$\"\"$

Step 4:

Substitute the values of $\"\"$ , $\"\"$ $\"\"$ and $\"\"$ in $\"\"$ .

$\"\"$ .

General solution of the differential equation is $\"\"$ .

$\"\"$

Solution of the differential equation is $\"\"$ .

Solution:

Solution of the differential equation is $\"\"$ .