Question

Solve by variation of parameters:

A. y" + 6y' + 9y = e^(-3t)/(1 + t^2)

B. y" − y = e^t − e^-t.

Answer 1

Question-(A)

.

for homogeneous system, find roots

for one real root, complementary solution is

.

take derivative

now find Wronskian

find determinant

and we have

.

.

now perticular solution is

.

.

general solution is

.

.

.

Question-(B)

.

for homogeneous system, find roots

for 2 real roots, complementary solution is

.

take derivative

now find Wronskian

find determinant

and we have

.

.

now perticular solution is

.

general solution is

take