Solve (2y+3xy^2)dx+(x+2x^2y)dy=0

$$( 2 y + 3 x y ^ { 2 } ) d x + ( x + 2 x ^ { 2 } y ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=0\text{ or }y=-\frac{3}{5x}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2y+3xy^{2} by d.

$$\left(2yd+3xy^{2}d\right)x+\left(x+2x^{2}y\right)dy=0$$

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

$$2ydx+3y^{2}dx^{2}+\left(x+2x^{2}y\right)dy=0$$

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

$$2ydx+3y^{2}dx^{2}+\left(xd+2x^{2}yd\right)y=0$$

Use the distributive property to multiply xd+2x^{2}yd by y.

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

Combine 2ydx and xdy to get 3ydx.

$$3ydx+3y^{2}dx^{2}+2x^{2}dy^{2}=0$$

Combine 3y^{2}dx^{2} and 2x^{2}dy^{2} to get 5y^{2}dx^{2}.

$$3ydx+5y^{2}dx^{2}=0$$

Combine all terms containing d.

$$\left(3yx+5y^{2}x^{2}\right)d=0$$

The equation is in standard form.

$$\left(5x^{2}y^{2}+3xy\right)d=0$$

Divide 0 by 3yx+5y^{2}x^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=0\text{ or }y=-\frac{3}{5x}\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2y+3xy^{2} by d.

$$\left(2yd+3xy^{2}d\right)x+\left(x+2x^{2}y\right)dy=0$$

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

$$2ydx+3y^{2}dx^{2}+\left(x+2x^{2}y\right)dy=0$$

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

$$2ydx+3y^{2}dx^{2}+\left(xd+2x^{2}yd\right)y=0$$

Use the distributive property to multiply xd+2x^{2}yd by y.

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

Combine 2ydx and xdy to get 3ydx.

$$3ydx+3y^{2}dx^{2}+2x^{2}dy^{2}=0$$

Combine 3y^{2}dx^{2} and 2x^{2}dy^{2} to get 5y^{2}dx^{2}.

$$3ydx+5y^{2}dx^{2}=0$$

Combine all terms containing d.

$$\left(3yx+5y^{2}x^{2}\right)d=0$$

The equation is in standard form.

$$\left(5x^{2}y^{2}+3xy\right)d=0$$

Divide 0 by 3yx+5y^{2}x^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\frac{3}{5y}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=-\frac{3}{5y}\text{, }&y\neq 0\\x\in \mathrm{R}\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$