# How can you solve this PDE analytically?

I have the PDE listed below and I am not quite sure how to solve it.

$$\frac{\partial^2 g}{\partial y^2} - \frac{\partial^2 g}{\partial z^2} - 2 \frac{\partial g}{\partial x}=0$$

I have tried separation of variables, however, it does not seem to work. I have been reading about the method of characteristics but it mostly applies to first order PDE. I am unfamiliar with that approach and I am not sure how I may apply it to this problem.

I am more interested in the process of how to solve this problem than the solution but I would gladly accept the solution as well so I can work it out for myself.

I appreciate any assistance.

• The easiest way is taking a guess as to the structure of $g$, and trying to see if that structure hosts a solution. Other methods are ... not fun. Check out this link. It might give you some ideas as to where to start. – Don Thousand Mar 24 at 22:57

## 1 Answer

Hint

The method "Separation of Variables" works. Define$$g(x,y,z)=f_1(x)f_2(y)f_3(z)$$therefore $$f_1(x)f_2''(y)f_3(z)-f_1(x)f_2(y)f_3''(z)-2f_1'(x)f_2(y)f_3(z)=0$$which yields to$${f_2''(y)\over f_2(y)}-{f_3''(z)\over f_3(z)}-2{f_1'(x)\over f_1(x)}=0$$and hence $${f_2''(y)\over f_2(y)}=k_1\\{f_3''(z)\over f_3(z)}=k_2\\{f_1'(x)\over f_1(x)}={k_1-k_2\over 2}$$

• Excellent. Thank you. – dsmalenb Mar 24 at 23:05
• Your welcome. Good luck! – Mostafa Ayaz Mar 24 at 23:08