APM346-2021S > Quiz 1

LEC9101 Quiz#1 B Question#1

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Xun Zheng:
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.

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