I am studying PDEs using the book "PDEs An Introduction 2nd edition" by Walter A. Strauss. In Chapter 2, a "geometric method" is described in order to solve linear PDEs of the type:
$$ (x,y)\mapsto u_x + yu_y = 0 $$
This is said to be equivalent to the directional derivative of $u$ in the direction of the vector $(1,y)$ being set to 0. Then characteristic curves having as tangent vectors $(1,y)$ are found
$$\frac{dy}{dx} = \frac{y}{1} \implies y = Ce^x $$
Since $u(x,y)$ is constant on these curves:
$$u(x,y) = f(e^{-x}y)$$
is the general solutions of the PDE, where $f$ is an arbitrary function. Now the same method is applied to solve more general equations, such as:
$$u_x + u_y + u = f(x,y)$$
I have tried to use the same method to solve the following differential equation:
$$yu_x -xu_y = 3x$$
where $ u(x,0) = x^2$
Then applying the method:
$$ \frac{dy}{dx} = \frac{-x}{y} \implies C = x^2 + y^2$$
The PDE reduces to an ODE:
$$\frac{du}{dx} = 3x/y \implies u (x,y) = 3x^2/2y + f(C)$$
However with the boundary imposed ($u(x,0) = x^2$) this seem impossible to solve since to find the particular solution I would have to divide the a term by zero. Does the method described in the book have limited scope? How can I solve this differential equation? Also, why is $u(x,y)$ constant on the "characteristic curves"?
Best Answer
You can't have this: $$\frac{du}{dx} = 3x/y ==> u (x,y) = 3x^2/y + f(C)$$ Because you have a differential equation ( du,dx) but three variables namely x,y,u. Y shouldnt be there...Apart for that mistake you did a very good job.
Usisng the method of characteristics:
$$\frac {dx}{y}=\frac {dy}{-x}=\frac {dz}{3x}$$ $$-xdx=ydy \implies x^2+y^2=K_1$$ $$3xdy=-xdz \implies y+z/3 =K_2$$ $$f(x^2+y^2)=y+u/3$$ $$u(x,y)=3f(x^2+y^2)-3y$$ we have $u(x,0)=x^2$ $$f(x^2)=x^2/3 \implies f(x)=x/3$$ Therefore $$\boxed{u(x,y)=x^2+y^2-3y}$$