Solve xe^xdy=(x^3+2ye^x)dx

$$x e ^ { x } d y = ( x ^ { 3 } + 2 y e ^ { x } ) d x$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&y=-\frac{x^{3}}{e^{x}}\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{3}+2ye^{x} by d.

$$xe^{x}dy=\left(x^{3}d+2ye^{x}d\right)x$$

Use the distributive property to multiply x^{3}d+2ye^{x}d by x.

$$xe^{x}dy=dx^{4}+2ye^{x}dx$$

Subtract dx^{4} from both sides.

$$xe^{x}dy-dx^{4}=2ye^{x}dx$$

Subtract 2ye^{x}dx from both sides.

$$xe^{x}dy-dx^{4}-2ye^{x}dx=0$$

Combine xe^{x}dy and -2ye^{x}dx to get -xe^{x}dy.

$$-xe^{x}dy-dx^{4}=0$$

Combine all terms containing d.

$$\left(-xe^{x}y-x^{4}\right)d=0$$

The equation is in standard form.

$$\left(-x^{4}-xye^{x}\right)d=0$$

Divide 0 by -xe^{x}y-x^{4}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=-\frac{x^{3}}{e^{x}}\text{ or }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{3}+2ye^{x} by d.

$$xe^{x}dy=\left(x^{3}d+2ye^{x}d\right)x$$

Use the distributive property to multiply x^{3}d+2ye^{x}d by x.

$$xe^{x}dy=dx^{4}+2ye^{x}dx$$

Subtract dx^{4} from both sides.

$$xe^{x}dy-dx^{4}=2ye^{x}dx$$

Subtract 2ye^{x}dx from both sides.

$$xe^{x}dy-dx^{4}-2ye^{x}dx=0$$

Combine xe^{x}dy and -2ye^{x}dx to get -xe^{x}dy.

$$-xe^{x}dy-dx^{4}=0$$

Combine all terms containing d.

$$\left(-xe^{x}y-x^{4}\right)d=0$$

The equation is in standard form.

$$\left(-x^{4}-xye^{x}\right)d=0$$

Divide 0 by -xe^{x}y-x^{4}.

$$d=0$$