Solve y=x^p+bxy

$$y = x ^ { p } + b x y$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{y-x^{p}}{xy}\text{, }&y\neq 0\text{ and }x\neq 0\\b\in \mathrm{C}\text{, }&y=0\text{ and }x=0\text{ and }p\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{p}+bxy=y$$

Subtract x^{p} from both sides.

$$bxy=y-x^{p}$$

The equation is in standard form.

$$xyb=y-x^{p}$$

Divide both sides by xy.

$$\frac{xyb}{xy}=\frac{y-x^{p}}{xy}$$

Dividing by xy undoes the multiplication by xy.

$$b=\frac{y-x^{p}}{xy}$$

Solve for b

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

Steps for Solving Linear Equation

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

$$x^{p}+bxy=y$$

Subtract x^{p} from both sides.

$$bxy=y-x^{p}$$

The equation is in standard form.

$$xyb=y-x^{p}$$

Divide both sides by xy.

$$\frac{xyb}{xy}=\frac{y-x^{p}}{xy}$$

Dividing by xy undoes the multiplication by xy.

$$b=\frac{y-x^{p}}{xy}$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\log_{x}\left(y\left(1-bx\right)\right)+\frac{2i\pi n_{1}}{\ln(x)}\text{, }n_{1}\in \mathrm{Z}\text{, }&\left(b=0\text{ or }x\neq \frac{1}{b}\right)\text{ and }y\neq 0\text{ and }x\neq 1\text{ and }x\neq 0\\p\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(x=1\text{ and }y=1\text{ and }b=0\right)\text{ or }\left(x=1\text{ and }y=\frac{1}{1-b}\text{ and }b\neq 1\right)\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\log_{x}\left(y\left(1-bx\right)\right)\text{, }&\left(y>0\text{ and }x<\frac{1}{b}\text{ and }x\neq 1\text{ and }x>0\text{ and }b>0\right)\text{ or }\left(y>0\text{ and }b=0\text{ and }x\neq 1\text{ and }x>0\right)\text{ or }\left(y>0\text{ and }x>\frac{1}{b}\text{ and }b<0\text{ and }x\neq 1\text{ and }x>0\right)\text{ or }\left(y<0\text{ and }x>\frac{1}{b}\text{ and }b>0\text{ and }x\neq 1\text{ and }x>0\right)\text{ or }\left(y<0\text{ and }x<\frac{1}{b}\text{ and }b<0\text{ and }x\neq 1\text{ and }x>0\right)\\p\in \mathrm{R}\text{, }&\left(x=1\text{ and }y=1\text{ and }b=0\right)\text{ or }\left(x=1\text{ and }y=\frac{1}{1-b}\text{ and }b\neq 1\right)\text{ or }\left(x=-1\text{ and }y=-1\text{ and }b=0\text{ and }Denominator(p)\text{bmod}2=1\text{ and }Numerator(p)\text{bmod}2=1\right)\text{ or }\left(x=-1\text{ and }y=-\frac{1}{b+1}\text{ and }b\neq -1\text{ and }Denominator(p)\text{bmod}2=1\text{ and }Numerator(p)\text{bmod}2=1\right)\\p>0\text{, }&x=0\text{ and }y=0\end{matrix}\right.$$