Solve dy/dx=ky(y-d)

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

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply ky by y-d.

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

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

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

Subtract ky^{2} from both sides.

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

Reorder the terms.

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

The equation is in standard form.

$$\left(-ky\right)d=-ky^{2}$$

Divide both sides by -ky.

$$\frac{\left(-ky\right)d}{-ky}=-\frac{ky^{2}}{-ky}$$

Dividing by -ky undoes the multiplication by -ky.

$$d=-\frac{ky^{2}}{-ky}$$

Divide -ky^{2} by -ky.

$$d=y$$

Solve for k

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

Steps for Solving Linear Equation

Use the distributive property to multiply ky by y-d.

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

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

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

Combine all terms containing k.

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

The equation is in standard form.

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

Divide 0 by y^{2}-yd.

$$k=0$$