Solve yydy=2x^2+3xy+2y^2

$$y y d y = 2 x ^ { 2 } + 3 x y + 2 y ^ { 2 }$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}\text{, }&y\neq 0\\d\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$y^{2}dy=2x^{2}+3xy+2y^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}d=2x^{2}+3xy+2y^{2}$$

Divide both sides by y^{3}.

$$\frac{y^{3}d}{y^{3}}=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}$$

Dividing by y^{3} undoes the multiplication by y^{3}.

$$d=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}\text{, }&y\neq 0\\d\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply y and y to get y^{2}.

$$y^{2}dy=2x^{2}+3xy+2y^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$y^{3}d=2x^{2}+3xy+2y^{2}$$

Divide both sides by y^{3}.

$$\frac{y^{3}d}{y^{3}}=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}$$

Dividing by y^{3} undoes the multiplication by y^{3}.

$$d=\frac{2x^{2}+3xy+2y^{2}}{y^{3}}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{y^{2}\left(8dy-7\right)}-3y}{4}$$

$$x=\frac{-\sqrt{y^{2}\left(8dy-7\right)}-3y}{4}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{y\left(\sqrt{8dy-7}-3\right)}{4}\text{; }x=\frac{y\left(-\sqrt{8dy-7}-3\right)}{4}\text{, }&\left(d<0\text{ or }y\geq \frac{7}{8d}\right)\text{ and }\left(d>0\text{ or }y\leq \frac{7}{8d}\right)\text{ and }d\neq 0\\x=0\text{, }&y=0\end{matrix}\right.$$