Solve dS/dt=-betaSI/N-vS

$$\frac { d S } { d t } = - \beta \frac { S I } { N } - v S$$

Solve for I

$$\left\{\begin{matrix}I=-\frac{Nv}{\beta }\text{, }&\beta \neq 0\text{ and }S\neq 0\text{ and }N\neq 0\\I\in \mathrm{R}\text{, }&\left(S=0\text{ and }N\neq 0\right)\text{ or }\left(v=0\text{ and }\beta =0\text{ and }N\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by N.

$$N\frac{\mathrm{d}}{\mathrm{d}t}(S)=\left(-\beta \right)SI-vSN$$

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

$$\left(-\beta \right)SI-vSN=N\frac{\mathrm{d}}{\mathrm{d}t}(S)$$

Add vSN to both sides.

$$\left(-\beta \right)SI=N\frac{\mathrm{d}}{\mathrm{d}t}(S)+vSN$$

Reorder the terms.

$$-IS\beta =N\frac{\mathrm{d}}{\mathrm{d}t}(S)+NSv$$

The equation is in standard form.

$$\left(-S\beta \right)I=NSv$$

Divide both sides by -S\beta .

$$\frac{\left(-S\beta \right)I}{-S\beta }=\frac{NSv}{-S\beta }$$

Dividing by -S\beta undoes the multiplication by -S\beta .

$$I=\frac{NSv}{-S\beta }$$

Divide NSv by -S\beta .

$$I=-\frac{Nv}{\beta }$$

Solve for N

$$\left\{\begin{matrix}N=-\frac{I\beta }{v}\text{, }&I\neq 0\text{ and }\beta \neq 0\text{ and }v\neq 0\text{ and }S\neq 0\\N\neq 0\text{, }&\left(\beta =0\text{ and }v=0\right)\text{ or }\left(I=0\text{ and }v=0\right)\text{ or }S=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$N\frac{\mathrm{d}}{\mathrm{d}t}(S)=\left(-\beta \right)SI-vSN$$

Add vSN to both sides.

$$N\frac{\mathrm{d}}{\mathrm{d}t}(S)+vSN=\left(-\beta \right)SI$$

Reorder the terms.

$$N\frac{\mathrm{d}}{\mathrm{d}t}(S)+NSv=-IS\beta$$

Combine all terms containing N.

$$\left(\frac{\mathrm{d}}{\mathrm{d}t}(S)+Sv\right)N=-IS\beta$$

The equation is in standard form.

$$SvN=-IS\beta$$

Divide both sides by Sv.

$$\frac{SvN}{Sv}=-\frac{IS\beta }{Sv}$$

Dividing by Sv undoes the multiplication by Sv.

$$N=-\frac{IS\beta }{Sv}$$

Divide -IS\beta by Sv.

$$N=-\frac{I\beta }{v}$$

Variable N cannot be equal to 0.

$$N=-\frac{I\beta }{v}\text{, }N\neq 0$$