Solve dy/dx=-ky(y-a)

$$\frac { d y } { d x } = - k y ( y - a )$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply \left(-k\right)y by y-a.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=\left(-k\right)y^{2}-\left(-k\right)ya$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=\left(-k\right)y^{2}+kya$$

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

$$\left(-k\right)y^{2}+kya=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Subtract \left(-k\right)y^{2} from both sides.

$$kya=\frac{\mathrm{d}}{\mathrm{d}x}(y)-\left(-k\right)y^{2}$$

Multiply -1 and -1 to get 1.

$$kya=\frac{\mathrm{d}}{\mathrm{d}x}(y)+ky^{2}$$

The equation is in standard form.

$$kya=ky^{2}$$

Divide both sides by ky.

$$\frac{kya}{ky}=\frac{ky^{2}}{ky}$$

Dividing by ky undoes the multiplication by ky.

$$a=\frac{ky^{2}}{ky}$$

Divide ky^{2} by ky.

$$a=y$$

Solve for k

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

Steps for Solving Linear Equation

Use the distributive property to multiply \left(-k\right)y by y-a.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=\left(-k\right)y^{2}-\left(-k\right)ya$$

Multiply -1 and -1 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(y)=\left(-k\right)y^{2}+kya$$

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

$$\left(-k\right)y^{2}+kya=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Reorder the terms.

$$-ky^{2}+aky=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Combine all terms containing k.

$$\left(-y^{2}+ay\right)k=\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

The equation is in standard form.

$$\left(ay-y^{2}\right)k=0$$

Divide 0 by -y^{2}+ay.

$$k=0$$