I want to show that the following PDE of the function $u(x,y)$ has infinitely many solutions:
$$
\left\{
\begin{array}{c}
u_x+u_y=2xu \\
u(x,x)=e^{x^2} \\
\end{array}
\right.
$$
By using the method of characteristics and choosing a curve $\Gamma(r,r,e^{r^2})$ in $u(x(r,s),y(r,s))$, I get the characteristic curve $(x(r,s),y(r,s),z(r,s))=(s+r,s+r,e^{(s+r)^2})$
I notice: $x(r,s)=y(r,s)$ $\implies$ $u(x,y)=z(r(x,y),s(x,y))=e^{x^2}=e^{y^2}$. However, this solution that I find is unique. I seem to be missing something to show that there are infinite solutions. Any tips?
Best Answer
$$u_x+u_y=2xu $$ The characteristic curves are solution of the differential equations : $$\frac{dx}{1}=\frac{dy}{1}=\frac{du}{2xu}$$ From $dx=dy$ a first family of characteristic curves is $\quad y-x=c_1$
From $dx=\frac{du}{2xu}$ a second family of characteristic curves is $\quad ue^{-x^2}=c_2$
The general solution of the PDE expressed on the form of implicit equation is : $$\Phi\left((y-x)\:,\:(ue^{-x^2})\right)=0$$ or, on explicit form : $$ue^{-x^2}=f(y-x) \quad\to\quad u=e^{x^2}f(y-x)$$ where $f$ is any differentiable function.
With the condition $u(x,x)=e^{x^2}=e^{x^2}f(x-x)=e^{x^2}f(0)\quad\implies\quad f(0)=1$
The solutions are : $$u(x,y)=e^{x^2}f(y-x)\quad \text{any function }f \text{ having the property }f(0)=1$$ Since they are an infinity of functions which have the property $f(0)=1$, this proves that they are an infinity of solutions for the PDE with condition $\begin{cases} u_x+u_y=2xu \\ u(x,x)=e^{x^2} \end{cases} $
EXAMPLE of solutions :
With $f(X)=C\quad$ a set of solutions is : $\quad u(x,y)=C\:e^{x^2}$
With $f(X)=CX\quad$ a set of solutions is : $\quad u(x,y)=C\:e^{x^2}(y-x)$
With $f(X)=CX^b\quad$ a set of solutions is : $\quad u(x,y)=C\:e^{x^2}(y-x)^b$
With $f(X)=C\sin(X)\quad$ a set of solutions is : $\quad u(x,y)=C\:e^{x^2}\sin(y-x)$
With $f(X)=Ce^{-bX^2}\quad$ a set of solutions is : $\quad u(x,y)=C\:e^{x^2}e^{-b(y-x)^2}$
An so on ...
One see that they are an infinity of examples, many are easy to find. And all linear combinations of those solutions.