# Solve the differential equation: \frac{dy}{dx}=\frac{x}{y}\sqrt{3+x^2}

## Question:

Solve the differential equation: {eq}\frac{dy}{dx}=\frac{x}{y}\sqrt{3+x^2}{/eq}

## Solution of the Differential Equation :

To find the solution of the differential equation we use the method of separation of variables. We can solve this by separating the equation into two parts. We move all of the equation involving the {eq}y{/eq} variable to one side and all of the equation involving the {eq}x{/eq} variable to the other side, then we can integrate both sides.

{eq}\displaystyle\int P(y)\: dy=\int Q(x)\: dx {/eq}

Consider the differential equation

{eq}\displaystyle\frac{dy}{dx}=\frac{x}{y}\sqrt{3+x^2} {/eq}

Seprate the variables of {eq}x {/eq} and {eq}y {/eq} and then integrate we get

{eq}\displaystyle\int y\: dy=\int x\sqrt{3+x^2}\: dx\\ \displaystyle\frac{y^2}{2}=\frac{\left(\sqrt{3+x^2}\right)^{2+1}}{2+1}+c\\ \displaystyle\frac{y^2}{2}=\frac{1}{3}\left(3+x^2\right)^{\frac{3}{2}}+c {/eq}

{eq}\displaystyle y^2=\frac{2}{3}\left(3+x^2\right)^{\frac{3}{2}}+C,\quad (2c=C {/eq} new constant)