Question

Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters.

Answer 1

(a)

Step 1:

The differential equation is .

The differential equation is in the form of .

is called complementary equation.

The general solution of is .

The auxiliary equation is .

and .

The roots of auxiliary equation is real and equal.

The solution of complementary equation is .

Step 2:

Take .

The is exponential function and continuous for all values of .

The general solution of is .

Substitute , and in .

.

Substitute in the general solution of .

.

The solution of differential equation is .

Substitute and .

.

Solution:

.

Answer 2

(b)

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 complementary solution 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:

The differential equation is in the form of .

is called complementary equation.

The general solution of is .

The auxiliary equation is .

and .

The roots of auxiliary equation is real and equal.

The solution of complementary equation is .

The general solution of is .

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

Find wronskian of and is

Answer 3

Contd...

Step 3:

Find .

Find .

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 .