# Find the general solution to the homogeneous differential equation...

## Question:

Find the general solution to the homogeneous differential equation {eq}\frac{d^2y}{dt^2}-15\frac{dy}{dt}=0 {/eq}.

## Homogeneous Differential Equation

Consider the following homogeneous second-order differential equation

{eq}\displaystyle f(t)y'' + g(t)y' = 0 \quad (*) {/eq}

Then, equation (*) can be solved by making the following substitutions

{eq}\displaystyle u = y' \Rightarrow u' = y'' {/eq}

So that equation (*) becomes the following separable differential equation

{eq}\displaystyle f(t)u' + g(t)u = 0 {/eq}

## Answer and Explanation: 1

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{eq}\displaystyle v = \frac{dy}{dt} \quad (1) \\ \\ \displaystyle \Rightarrow \frac{dv}{dt} = \frac{d^2y}{dt^2} \quad (2) {/eq}

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Chapter 16 / Lesson 1Discover what separable differential equations are and their uses. Learn to identify if an equation is separable and how to solve them through given examples.

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