Question

Solve nonhomogenous equation: x"+4x=cost

Answer 1

`x(t) = Acos 2t + B sin 2t+(cost)/(3)`

Explanation:

Given is a non homogeneous second degree equation as

`x`

Auxialary equation is

`m^2+4 =0\\m = 2i, -2i`

Hence general solution is

x = Acos 2t + B sin 2t

Particular integral is =
`(cost)/(D^2+4)`

Since t has coefficient 1, we substitute

`D^2 =-1\\PI = (cost)/(-1+4) =(cost)/(3)`

Hence full solution is

`x(t) = Acos 2t + B sin 2t+(cost)/(3)`