Solve y^prime=ay-bxy

$$y ^ { \prime } = a y - b x y$$

Solve for a (complex solution)

$$\left\{\begin{matrix}\\a=bx\text{, }&\text{unconditionally}\\a\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ay-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Add bxy to both sides.

$$ay=\frac{\mathrm{d}}{\mathrm{d}x}(y)+bxy$$

The equation is in standard form.

$$ya=bxy$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{bxy}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{bxy}{y}$$

Divide bxy by y.

$$a=bx$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{a}{x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&y=0\text{ or }\left(a=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ay-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract ay from both sides.

$$-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)-ay$$

The equation is in standard form.

$$\left(-xy\right)b=-ay$$

Divide both sides by -xy.

$$\frac{\left(-xy\right)b}{-xy}=-\frac{ay}{-xy}$$

Dividing by -xy undoes the multiplication by -xy.

$$b=-\frac{ay}{-xy}$$

Divide -ay by -xy.

$$b=\frac{a}{x}$$

Solve for a

$$\left\{\begin{matrix}\\a=bx\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ay-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Add bxy to both sides.

$$ay=\frac{\mathrm{d}}{\mathrm{d}x}(y)+bxy$$

The equation is in standard form.

$$ya=bxy$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{bxy}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{bxy}{y}$$

Divide bxy by y.

$$a=bx$$

Solve for b

$$\left\{\begin{matrix}b=\frac{a}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&y=0\text{ or }\left(a=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$ay-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract ay from both sides.

$$-bxy=\frac{\mathrm{d}}{\mathrm{d}x}(y)-ay$$

The equation is in standard form.

$$\left(-xy\right)b=-ay$$

Divide both sides by -xy.

$$\frac{\left(-xy\right)b}{-xy}=-\frac{ay}{-xy}$$

Dividing by -xy undoes the multiplication by -xy.

$$b=-\frac{ay}{-xy}$$

Divide -ay by -xy.

$$b=\frac{a}{x}$$