First Order ODE/exp x (1 + x) dx = (x exp x - y exp y) dy

Theorem
The first order ordinary differential equation:


 * $(1): \quad e^x \paren {1 + x} \rd x = \paren {x e^x - y e^y} \rd y$

has the solution:


 * $2 x e^x e^{-y} + y^2 + C$

Proof
Dividing $(1)$ through by $e^y$:
 * $(2): \quad \paren {1 + x} e^x e^{-y} \rd x = \paren {x e^x e^{-y} - y} \rd y$

Let $z = x e^x e^{-y}$.

Then:
 * $\dfrac {\d z} {\d x} = \paren {1 + x} e^x e^{-y} - x e^x e^{-y} \dfrac {\d y} {\d x}$

or:
 * $\d z = \paren {1 + x} e^x e^{-y} \rd x - x e^x e^{-y} \rd y$

Substituting $z$ and $\d z$ in $(2)$:
 * $\d z = \paren {z - y} \rd y - z \rd y = - y \rd y$

which directly leads to:
 * $\ds \int \rd z = -\int y \rd y$

from which:
 * $z = -\dfrac {y^2} 2 + k$

Substituting for $z$ and letting $C = 2 k$:
 * $2 x e^x e^{-y} + y^2 + C$