Solve f^prime(x)=-b/mf(x)-g

$$f ^ { \prime } ( x ) = - \frac { b } { m } f ( x ) - g$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{gm}{fx}\text{, }&x\neq 0\text{ and }f\neq 0\text{ and }m\neq 0\\b\in \mathrm{R}\text{, }&\left(x=0\text{ or }f=0\right)\text{ and }g=0\text{ and }m\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by m.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xm=\left(-\frac{b}{m}\right)fxm-gm$$

Express \left(-\frac{b}{m}\right)f as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xm=\frac{-bf}{m}xm-gm$$

Express \frac{-bf}{m}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xm=\frac{-bfx}{m}m-gm$$

Express \frac{-bfx}{m}m as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xm=\frac{-bfxm}{m}-gm$$

Cancel out m in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(f)xm=-bfx-gm$$

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

$$-bfx-gm=\frac{\mathrm{d}}{\mathrm{d}x}(f)xm$$

Add gm to both sides.

$$-bfx=\frac{\mathrm{d}}{\mathrm{d}x}(f)xm+gm$$

The equation is in standard form.

$$\left(-fx\right)b=gm$$

Divide both sides by -fx.

$$\frac{\left(-fx\right)b}{-fx}=\frac{gm}{-fx}$$

Dividing by -fx undoes the multiplication by -fx.

$$b=\frac{gm}{-fx}$$

Divide gm by -fx.

$$b=-\frac{gm}{fx}$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{gm}{bx}\text{, }&x\neq 0\text{ and }b\neq 0\text{ and }m\neq 0\\f\in \mathrm{R}\text{, }&\left(x=0\text{ or }b=0\right)\text{ and }g=0\text{ and }m\neq 0\end{matrix}\right.$$