2nd order linear hyperbolic PDE?

[kingwinner]

u_xx - x² u_yy = 0 (assume x>0)

Is there any systematic method (e.g. change of variables) to solve this hyperbolic equation?

dy/dx = [B + sqrt(B² - AC)]/A
=> dy/dx = x
=> 2y -x² = c

dy/dx = [B - sqrt(B² - AC)]/A
=> dy/dx = -x
=> 2y + x² = k

So the characteristic curves are 2y -x² = c and 2y + x² = k

Now does this imply that the general solution is u = f(2y-x²)+g(2y+x²) ?

Thanks!

 

[matematikawan]

u = f(2y-x2)+g(2y+x2)

Is u_xx - x² u_yy identically zero? Then it must be a general solution

 

[kingwinner]

Hi, I tried to verify it by computing a bunch of tedious second derivatives, but somehow I didn't get 0... I may have made a mistake in my calculations...

 

[kosovtsov]

The most direct way to solve this PDE is the Fourier method, that is, supposing that solution has the form

u(x,y)=\int_{-\infty}^{\infty}F(x,k)exp(iyk) dk

we obtain ODE for F

F_{xx}+x^2k^2F=0 ,

which solution is as follows

F(x,k)=x^{1/2}[F1(k)J_{1/4}(x^2k/2)+F2(k)K_{1/4}(x^2k/2)] ,

where J_{1/4} and K_{1/4} are modified Bessel functions; F1 and F2 are arbitrary functions.