Solve dy/dx+P(x)y=Q(x)y^n

$$\frac { d y } { d x } + P ( x ) y = Q ( x ) y ^ { n }$$

Solve for P

$$\left\{\begin{matrix}P=Qy^{n-1}\text{, }&x\neq 0\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }y>0\right)\text{ and }y\neq 0\\P\in \mathrm{R}\text{, }&\left(x=0\text{ and }y>0\right)\text{ or }\left(x=0\text{ and }y<0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(y=0\text{ and }n>0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \frac{\mathrm{d}}{\mathrm{d}x}(y) from both sides.

$$Pxy=Qxy^{n}-\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

The equation is in standard form.

$$xyP=Qxy^{n}$$

Divide both sides by xy.

$$\frac{xyP}{xy}=\frac{Qxy^{n}}{xy}$$

Dividing by xy undoes the multiplication by xy.

$$P=\frac{Qxy^{n}}{xy}$$

Divide Qxy^{n} by xy.

$$P=Qy^{n-1}$$

Solve for Q

$$\left\{\begin{matrix}Q=Py^{1-n}\text{, }&x\neq 0\text{ and }\left(Denominator(n)\text{bmod}2=1\text{ or }y>0\right)\text{ and }y\neq 0\\Q\in \mathrm{R}\text{, }&\left(x=0\text{ and }y>0\right)\text{ or }\left(x=0\text{ and }y<0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(y=0\text{ and }n>0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$Qxy^{n}=\frac{\mathrm{d}}{\mathrm{d}x}(y)+Pxy$$

The equation is in standard form.

$$xy^{n}Q=Pxy$$

Divide both sides by xy^{n}.

$$\frac{xy^{n}Q}{xy^{n}}=\frac{Pxy}{xy^{n}}$$

Dividing by xy^{n} undoes the multiplication by xy^{n}.

$$Q=\frac{Pxy}{xy^{n}}$$

Divide Pxy by xy^{n}.

$$Q=Py^{1-n}$$