In mathematics, Euler's differential equation is a first-order non-linear ordinary differential equation, named after Leonhard Euler. It is given by:
d y d x + a 0 + a 1 y + a 2 y 2 + a 3 y 3 + a 4 y 4 a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 = 0 {\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:
∫ d y a 0 + a 1 y + a 2 y 2 + a 3 y 3 + a 4 y 4 + ∫ d x a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 = c {\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}
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