# Toronto Math Forum

## APM346-2021S => APM346--Tests & Quizzes => Quiz 1 => Topic started by: Xun Zheng on January 28, 2021, 03:22:05 PM

Title: LEC9101 Quiz#1 B Question#1
Post by: Xun Zheng on January 28, 2021, 03:22:05 PM
Problem 1 (1.5 pt). Consider first order equations and determine if they are linear homogeneous, linear inhomogeneous, or nonlinear (u is an unknown function); for nonlinear equations, indicate if they are also semilinear, or quasilinear:
$$u_t+xu_x-u=0$$

Solution:

First, according to the equations of the form Lu=f(x), we have that f=0.

Then, $u_t+xu_x-u=0$ is a homogeneous equation.

Next, we can find the operator L.

Substituting (u+v) and (cu) to the L, we get
$$L(u+v)=L(u)+L(v) ,$$
$$L(cu)=cL(u).$$
Hence, $u_t+xu_x-u=0$ is linear.

Therefore, we get that $u_t+xu_x-u=0$ is a linear homogeneous equation.