# Find the general solution to 3y^{\prime\prime} + 108 y =0 . Give your answer as y = ... In your...

## Question:

Find the general solution to {eq}3y^{\prime\prime} + 108 y =0 {/eq}. Give your answer as y = ...

In your answer, use {eq}c_1 {/eq} and {eq}c_2 {/eq} to denote arbitrary constants and x the independent variable.

## Linear Homogeneous Differential Equation:

Recall that for the second-order homogeneous linear differential equation {eq}ay''+by'+cy=0 {/eq} we call the equation {eq}ar^2+br+c=0 {/eq} the auxiliary equation and there are three cases.

Case1: The auxiliary equation has distinct real roots {eq}r_1 {/eq} and {eq}r_2. {/eq}

Then the general solution is {eq}y_c=c_1e^{r_1x}+c_2e^{r_2x}. {/eq}

Case 2: The auxiliary equation has one repeated real roots {eq}r. {/eq}

Then the general solution is {eq}y_c=c_1e^{rx}+c_2xe^{rx}. {/eq}

Case 3: The auxiliary equation has a pair of complex solutions of the form {eq}\alpha + \beta x {/eq} and {eq}\alpha -\beta x. {/eq}

Then the general solution is {eq}y_c=e^{\alpha x}(c_1\sin \beta x+c_2\cos \beta x). {/eq}

The auxiliary equation for {eq}3y''+108y=0 {/eq} is

{eq}3r^2+108=0\\ 3(r^2+36)=0 {/eq}

This has complex roots of {eq}r=\pm 6i. {/eq}

Then the general solution is

{eq}y=C_1\sin (6x)+C_2\cos (6x). {/eq}