# Toronto Math Forum

## APM346-2016F => APM346--Tests => Q1 => Topic started by: Victor Ivrii on September 29, 2016, 09:29:03 PM

Title: Q1-P1
Post by: Victor Ivrii on September 29, 2016, 09:29:03 PM
Consider first order equations and determine if they are linear homogeneous, linear inhomogeneous, quasilinear or non-linear ($u$ is an unknown function):
\begin{align}
&u_t+xu_x-u= 0,\label{eq-1}\\[5pt]
&u_x^2+u_y^2-1= 0. \label{eq-2}
\end{align}
Title: Re: Q1-P1
Post by: Shentao YANG on September 29, 2016, 09:43:40 PM
Below is my solution:
$$u_t+xu_x-u= 0\text{ : linear homogeneous}$$
$$u_x^2+u_y^2-1= 0\text{ : nonlinear}$$
Title: Re: Q1-P1
Post by: John Menacherry on September 29, 2016, 10:16:46 PM
Aren't they both linear inhomogeneous?
Title: Re: Q1-P1
Post by: Jaisen on September 30, 2016, 10:03:37 AM
John, I think because of the minuses you are right they are both inhomogeneous. But (1) is Semi linear since F=u but for Linear F has to be a function of (x,y). As for (2) it is fully non-linear (not quasi-linear) because of the squares.
Title: Re: Q1-P1
Post by: Victor Ivrii on September 30, 2016, 10:12:46 AM
Shentao YANG
is correct, it is linear homogeneous since $f(x,t,u)= c(x,y)u$.

The second equation is non-linear