# Thread: Solve this nonlinear differential equation

1. ## Solve this nonlinear differential equation

Hi, I'd like to solve the following differential equation to find $y$ as a function of $x$ (in fact I know what the answer should be due to the context, but I have no idea how to go about actually solving it). Any ideas would be appreciated!

$1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{y^2}{c^2}$,

where $c$ is a constant.

2. ## Re: Solve this nonlinear differential equation

Originally Posted by Amanda1990
Hi, I'd like to solve the following differential equation to find $y$ as a function of $x$ (in fact I know what the answer should be due to the context, but I have no idea how to go about actually solving it). Any ideas would be appreciated!

$1 + \left(\frac{\text{d}y}{\text{d}x}\right)^2 = \frac{y^2}{c^2}$,

where $c$ is a constant.
\displaystyle \begin{align*} 1 + \left(\frac{dy}{dx}\right)^2 &= \frac{y^2}{c^2} \\ \left(\frac{dy}{dx}\right)^2 &= \frac{y^2}{c^2} - 1 \\ \left(\frac{dy}{dx}\right)^2 &= \frac{y^2 - c^2}{c^2} \\ \frac{dy}{dx} &= \frac{\pm\sqrt{y^2 - c^2}}{|c|} \\ \pm \frac{1}{\sqrt{y^2 - c^2}}\,\frac{dy}{dx} &= \frac{1}{|c|} \end{align*}

You should now be able to solve this (it's separable).

3. ## Re: Solve this nonlinear differential equation

Aha - thanks!