Consider the equation
$u_x + u_y= 0$
$u(s,s) = 1$
Does the solution exist? If so, is it unique?
$u_x + u_y= 0$
Using the method of characteristics, we get: $\frac{dx}1 = \frac{dy}1 = \frac{du}{0} \rightarrow u = c_1 \text{ and } y-x = c_2$
So the general solution will be $u = f(y-x)$ where $f$ is arbitrary
Now how do I use $u(s,s) = 1$. I don't understand what $s$ is here? I am reading Partial Differential Equations of Fritz John and in Chapter 1, he introduces this parameter $s$ to solve Quasilinear PDEs but I didn't understand his method. Can someone help me please
$u(s,s)$ denotes the particular case when $x=s$ and $y=s$ for any values of $s$.
Since the general solution is $u(x,y)=f(y-x)$ thus $$u(s,s)=f(s-s)=f(0)=1$$ This means that any function $f$ which as the property $f(0)=1$ satisfies the condition.
They are an infinity of functions such as $f(0)=1$. For examples $f(X)=X+1$ or $f(X)=(X+1)^n$ or $f(X)=\cos(X)$ or $f(X)=e^X$ or etc.
This proves that they are an infinity of solutions of the PDE with the condition. For examples : $u(x,y)=y-x+1$ or $u(x,y)=(y-x+1)^n$ or $u(x,y)=\cos(y-x)$ or $u(x,y)=e^{y-x}$ or etc.