Solve dy/dx=ky(a-y)

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

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 ky by a-y.

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

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

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

Add ky^{2} to both sides.

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

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 ky by a-y.

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

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

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

Combine all terms containing k.

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

The equation is in standard form.

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

Divide 0 by ya-y^{2}.

$$k=0$$