# Solve first order ODEs x\sqrt{y^{2}+9}dx+5x^{9}ydy=0

## Question:

Solve first order ODEs

{eq}x\sqrt{y^{2}+9}dx+5x^{9}ydy=0 {/eq}

## Variable separation method

Separable equations have dy/dx (or dy/dt) equal to some expression.

Whereas in substitution it doesn't appear like that.

In this method, we separate the same type of variables to one side and other types to another side.

Then we integrate on both sides thus making it easy to solve the particular differential equation.

Given function is {eq}\displaystyle x\sqrt{y^{2}+9}dx+5x^{9}ydy=0\\ {/eq}

On rearranging the variables,{eq}\displaystyle \begin{align} x\sqrt{y^{2}+9}dx &= - 5x^{9}ydy\\ \frac{xdx}{-5x^{9}} &= \frac{ydy}{\sqrt{y^{2}+9}}\\ -0.2x^{-8}dx &= 0.5 \frac{2ydy}{\sqrt{y^{2}+9}} \end{align} {/eq}

Now variables are separated , integrating on both sides,

{eq}\displaystyle \begin{align} \int -0.2x^{-8}dx &= \int 0.5 \frac{2ydy}{\sqrt{y^{2}+9}}\\ 0.028x^{-7} + c &= y^{2}+9\\ \end{align} {/eq}