Toronto Math Forum
APM3462021S => APM346Tests & Quizzes => Quiz 1 => Topic started 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:
$$u_t+xu_xu=0$$
Solution:
First, according to the equations of the form Lu=f(x), we have that f=0.
Then, $u_t+xu_xu=0$ 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, $u_t+xu_xu=0$ is linear.
Therefore, we get that $u_t+xu_xu=0$ is a linear homogeneous equation.