Solve this nonlinear differential equation

**Amanda1990** · March 24th 2012, 03:07 AM

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.

**Prove It** · March 24th 2012, 03:46 AM

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).

**Amanda1990** · March 24th 2012, 04:03 AM

Aha - thanks!

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