Solve u_{t}+f(u)u_{x}=0

$$u _ { t } + f ( u ) u _ { x } = 0$$

Solve for f

$$\left\{\begin{matrix}f=-\frac{u_{t}}{uu_{x}}\text{, }&u_{x}\neq 0\text{ and }u\neq 0\\f\in \mathrm{R}\text{, }&\left(u_{x}=0\text{ or }u=0\right)\text{ and }u_{t}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract u_{t} from both sides. Anything subtracted from zero gives its negation.

$$fuu_{x}=-u_{t}$$

The equation is in standard form.

$$uu_{x}f=-u_{t}$$

Divide both sides by uu_{x}.

$$\frac{uu_{x}f}{uu_{x}}=-\frac{u_{t}}{uu_{x}}$$

Dividing by uu_{x} undoes the multiplication by uu_{x}.

$$f=-\frac{u_{t}}{uu_{x}}$$

Solve for u

$$\left\{\begin{matrix}u=-\frac{u_{t}}{fu_{x}}\text{, }&u_{x}\neq 0\text{ and }f\neq 0\\u\in \mathrm{R}\text{, }&\left(u_{x}=0\text{ or }f=0\right)\text{ and }u_{t}=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract u_{t} from both sides. Anything subtracted from zero gives its negation.

$$fuu_{x}=-u_{t}$$

The equation is in standard form.

$$fu_{x}u=-u_{t}$$

Divide both sides by fu_{x}.

$$\frac{fu_{x}u}{fu_{x}}=-\frac{u_{t}}{fu_{x}}$$

Dividing by fu_{x} undoes the multiplication by fu_{x}.

$$u=-\frac{u_{t}}{fu_{x}}$$