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: ^[1]

${\displaystyle {\frac {dy}{dx}}+{\frac {\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=0}$

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

${\displaystyle \int {\frac {dy}{\sqrt {a_{0}+a_{1}y+a_{2}y^{2}+a_{3}y^{3}+a_{4}y^{4}}}}+\int {\frac {dx}{\sqrt {a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}}}}=c}$

References

1. Ince, E. L. "L. 1944 Ordinary Differential Equations." 227.