[Math] Solution of $ydx-xdy+3x^2y^2e^{x^2}dx=0$

Question

Find the solution of given differential equation:

$$ydx-xdy+3x^2y^2e^{x^2}dx=0$$

I am not able to solve this because of $e^{x^2}$. Could someone help me with this one?

Answer 1

By dividing both side to $y^2$ we get $$ydx-xdy+3x^{ 2 }y^{ 2 }e^{ x^{ 2 } }dx=0\\ \frac { ydx-xdy }{ { y }^{ 2 } } +3x^{ 2 }e^{ x^{ 2 } }=0\\ d\left( \frac { x }{ y } \right) =3x^{ 2 }e^{ x^{ 2 } }\\ \int { d\left( \frac { x }{ y } \right) =\int { 3x^{ 2 }e^{ x^{ 2 } }dx } } =\frac { 3 }{ 2 } \int { x } d{ e }^{ { x }^{ 2 } }=\frac { 3 }{ 2 } \left( x{ e }^{ { x }^{ 2 } }-\int { { e }^{ { x }^{ 2 } }dx } \right) \\ \frac { x }{ y } =\frac { 3 }{ 2 } \left( x{ e }^{ { x }^{ 2 } }-\int { { e }^{ { x }^{ 2 } }dx } \right) +C\\ y=\frac { 2x }{ 3\left( \left( x{ e }^{ { x }^{ 2 } }-\int { { e }^{ { x }^{ 2 } }dx } \right) +C \right) } $$