# Find the general solution of the differential equation. x^2y'' + xy' + 4y = sin(2 ln x) cos(2 ln...

## Question:

Find the general solution of the differential equation.

{eq}x^2y'' + xy' + 4y = \sin(2 \ln x) \cos(2 \ln x), \ \ x > 0 {/eq}.

## Non-Homogeneous 2nd Order Linear ODEs:

Given a non-homogeneous 2nd Order Linear ODE of the form

{eq}a(x)y'' + b(x) y'+ c(x) y = f(x) {/eq}

have a solution that is composed of a homogeneous solution {eq}y_h {/eq} and a particular solution {eq}y_p {/eq}.

The homogeneous solution {eq}y_h {/eq} takes the form

{eq}y_h = c_1 y_1 + c_2 y_2 {/eq}

where {eq}c_1 {/eq} and {eq}c_2 {/eq} are constants and {eq}y_1 {/eq} and {eq}y_2 {/eq} are two independent solutions to the homogeneous problem:

{eq}a(x)y'' + b(x) y'+ c(x) y = 0 {/eq}.

The particular solution contains no undetermined constants and solves the original problem such that the general solution of the problem is

{eq}y = y_h + y_p {/eq}.