# Find the general solution of the given homogeneous differential equation. Verify your found...

## Question:

Find the general solution of the given homogeneous differential equation. Verify your found solution and show verification work.

{eq}\displaystyle 3 y'' + 2 y' + y = 0 {/eq}.

## Second Order Differential Equation.

The general solution of the homogeneous differential equation consists of only complementary function.

For finding complementary function, we write the auxiliary equation of the differential equation.

The solution depends on the roots of the auxiliary equation.

For complex roots of the auxiliary equation, the solution is :

{eq}y(t)=e^{\alpha t}\left (C_{1}\cos \beta t+C_{2}\sin\beta t \right )\\ {/eq}

where {eq}C_1, C_2 {/eq} are constants.

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Given:

{eq}3 y'' + 2 y' + y = 0 {/eq}

This is a second order differential equation.

Writing the auxiliary equation of the given differential...