# Third Order PDE written as a System of (Linear) First Order PDEs

I need to rewrite the PDE $$f_{y}+ff_{x}+f_{xxx}=0,$$ where $f=f(x,y)$ as a system of first order quasi-linear PDEs.

I have no idea how to tackle this problem. Any form of help will be appreciated. Are there any methodologies for such questions?

Thanks, Jay.

• You could rewrite it as a 3rd order nonlinear ode. The most general solution is $f=x^{-2}g(\mu)$ where $\mu=\frac{x^3}{y}$. If you take derivatives and substitute you get $$27\mu^3g^{(3)}+(24\mu+3\mu g-\mu^2)g'-2g^2-24g=0$$ Use numerical techniques to find $g$ and you're done. I know you weren't asked to solve the equation, but this approach (dreamed up by American mathematician Garrett Birkhoff) is a dandy approach when you've got to have an answer. – atomteori Jul 18 '14 at 13:10
• Dear @atomteori, Thanks for the additional approach. Very interesting. Can you direct me to an appropriate link to learn more about this? In particular, how were the forms of $f$ and $\mu$ selected. – Radz Jul 18 '14 at 13:28
• This is an application of Lie theory. The first task is to find a Lie group invariant to your DEQ, in this case $G(x,y,f)=(\lambda x, – atomteori Jul 18 '14 at 15:02 • Sorry, didn't get to finish last comment. Here's the rest.$G(x,y,f)=(\lambda x,\lambda^3 y,\lambda^{-2} f)\lambda_o=1$Now find a couple of non-trivial stabilizers for the group, such as$\mu=\frac{x^3}{y}$and$\nu=fx^2$, set$\nu=g(\mu)$and you're off to the races. A good reference is Lawrence Dresner's Similarity Solutions of Nonlinear Partial Differential Equations, ISBN 0-273-08621-9 It's probably out of print but you can find it on Amazon or Ebay, I think. Haven't checked recently. – atomteori Jul 18 '14 at 15:11 ## 1 Answer Hint:$g=f_x$so$f_{xx}=g_x$. Now$h=g_x$so$f_{xxx}=?$. Just add more dependent variables (unknown functions) • Dear @Sergio Parreiras, Thanks. So,$ f_{xxx}=g_{xx}=h_{x} $. Question: How do I write the aforementioned substitutions together with the (now reduced) original PDE as a system? – Jay Jul 18 '14 at 11:58 • You will have additional equations, just make sure every term in each equation has first or zero order. The first equation$g-f_x=0$and also the third I wrote should be in the new system but not the second. – Sergio Parreiras Jul 18 '14 at 12:06 • Dear @Sergio Parreiras, Thanks. So, I believe one more equation is required to complete the system. Last one is$f_{y}+fg+h_{x}=0 \$? – Jay Jul 18 '14 at 12:17
• Yes that is correct. Unrelated: do you know what is an integrability condition for a PDE? – Sergio Parreiras Jul 18 '14 at 12:22
• Thanks. I don't think so. However, I know that the above equation has a reduction to a Painleve equation, which is an integrable ODE. – Jay Jul 18 '14 at 12:34