Solutions of  :

The auxiliary equation is .

1. If roots of the auxiliary equation are real and distinct, then the general solution is

Step 1 :

The differential equation is and the initial conditions are .

The auxiliary equation is .

Find roots of the auxiliary equation.

The roots of the auxiliary equation .

The general solution is

Step 2 :

Substitute the initial condition in equation (1)

Differentiate equation (1) with respect to x.

Substitute the initial condition in above equation.

Step 3 :

Subtract equation (2) from equation (3).

Substitute in equation (2).

Substitute and in equation (1).

The general solution is .

Solution :

The general solution is .