Solve ay/dt=0.3y(1-y/0.06)

$$\frac { a y } { d t } = 0.3 y ( 1 - \frac { y } { 0.06 } )$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{dt\left(50y-3\right)}{10}\text{, }&d\neq 0\text{ and }t\neq 0\\a\in \mathrm{R}\text{, }&y=0\text{ and }d\neq 0\text{ and }t\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by dt.

$$ay=0.3y\left(1-\frac{y}{0.06}\right)dt$$

Use the distributive property to multiply 0.3y by 1-\frac{y}{0.06}.

$$ay=\left(0.3y+0.3y\left(-\frac{y}{0.06}\right)\right)dt$$

Use the distributive property to multiply 0.3y+0.3y\left(-\frac{y}{0.06}\right) by d.

$$ay=\left(0.3yd+0.3y\left(-\frac{y}{0.06}\right)d\right)t$$

Use the distributive property to multiply 0.3yd+0.3y\left(-\frac{y}{0.06}\right)d by t.

$$ay=0.3ydt+0.3y\left(-\frac{y}{0.06}\right)dt$$

Multiply 0.3 and -1 to get -0.3.

$$ay=0.3ydt-0.3y\times \frac{y}{0.06}dt$$

The equation is in standard form.

$$ya=-5dty^{2}+\frac{3dty}{10}$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{dty\left(0.3-5y\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{dty\left(0.3-5y\right)}{y}$$

Divide ydt\left(0.3-5y\right) by y.

$$a=dt\left(0.3-5y\right)$$

Solve for d

$$\left\{\begin{matrix}\\d\neq 0\text{, }&\text{unconditionally}\\d=-\frac{10a}{t\left(50y-3\right)}\text{, }&a\neq 0\text{ and }y\neq \frac{3}{50}\text{ and }t\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dt.

$$ay=0.3y\left(1-\frac{y}{0.06}\right)dt$$

Use the distributive property to multiply 0.3y by 1-\frac{y}{0.06}.

$$ay=\left(0.3y+0.3y\left(-\frac{y}{0.06}\right)\right)dt$$

Use the distributive property to multiply 0.3y+0.3y\left(-\frac{y}{0.06}\right) by d.

$$ay=\left(0.3yd+0.3y\left(-\frac{y}{0.06}\right)d\right)t$$

Use the distributive property to multiply 0.3yd+0.3y\left(-\frac{y}{0.06}\right)d by t.

$$ay=0.3ydt+0.3y\left(-\frac{y}{0.06}\right)dt$$

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

$$0.3ydt+0.3y\left(-\frac{y}{0.06}\right)dt=ay$$

Multiply 0.3 and -1 to get -0.3.

$$0.3ydt-0.3y\times \frac{y}{0.06}dt=ay$$

Combine all terms containing d.

$$\left(0.3yt-0.3y\times \frac{y}{0.06}t\right)d=ay$$

The equation is in standard form.

$$\left(-5ty^{2}+\frac{3ty}{10}\right)d=ay$$

Divide both sides by 0.3yt-5ty^{2}.

$$\frac{\left(-5ty^{2}+\frac{3ty}{10}\right)d}{-5ty^{2}+\frac{3ty}{10}}=\frac{ay}{-5ty^{2}+\frac{3ty}{10}}$$

Dividing by 0.3yt-5ty^{2} undoes the multiplication by 0.3yt-5ty^{2}.

$$d=\frac{ay}{-5ty^{2}+\frac{3ty}{10}}$$

Divide ay by 0.3yt-5ty^{2}.

$$d=\frac{a}{t\left(0.3-5y\right)}$$

Variable d cannot be equal to 0.

$$d=\frac{a}{t\left(0.3-5y\right)}\text{, }d\neq 0$$