Euler's differential equation

For other differential equations named after Euler, see List of things named after Leonhard Euler § Equations.

In mathematics, Euler's differential equation is a first-order non-linear ordinary differential equation, named after Leonhard Euler. It is given by:

$\frac{dy}{dx}+\frac{\sqrt{a_0+a_{1}y+a_2{y}^2+a_3{y}^3+a_4{y}^4}}{\sqrt{a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4}}=0$

This is a separable equation and the solution is given by the following integral equation:

$\int \frac{dy}{\sqrt{a_0+a_{1}y+a_2{y}^2+a_3{y}^3+a_4{y}^4}}+\int \frac{dx}{\sqrt{a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4}}=c$