So I'm working on a solution to this PDE,
I started by classifying the PDE and found it to be hyperbolic. So, I attempted to reduce it to canonical form, making the substitutions
Then, I obtained that
Substituting this back into the PDE, I end up with the equation
At about this point, I'm getting stuck on finding a general solution to the PDE by simplifying it further. I suspect the problem is with my substitution, specifically with , but I'm not really sure what's wrong with it, other than it looking a bit suspect. Can anyone point me in the right direction?
Thank you so much for that. It helped a lot to see that step worked out. Sorry for asking again, but could you possibly take a look at what I've done here and tell me where I'm going wrong? I'm sure it's something stupid.
I'm working on solving
Once again, the equation is hyperbolic so I found appopriate substitutions to be
So, taking the derivatives I find
Plugging these back into the pde, I end up with the equation (in canonical form),
I tried integrating with respect to eta to get
For some arbitrary B. However, I can't solve this ODE. I tried using an integrating factor, but that got messy quickly. I might have made an algebraic mistake in the reduction, but I've gotten the same canonical form twice.
If someone could take a look and show me where I went awry, I'd be most apprecitive.