Partial Differential Equations Question

[Hendrick]

Homework Statement

Find the link between constants $$\omega$$ and $$\beta$$

so that http://www4e.wolframalpha.com/Calculate/MSP/MSP181963g2e5f4i43d3b00005ief8e24920ah323?MSPStoreType=image/gif&s=20
is a solution of $$\frac{\partial^{2} u}{\partial x^{2}}=2\frac{\partial u}{\partial t}$$

(A & B are constants)

Homework Equations

I think that $$\frac{\partial^{2} u}{\partial x^{2}}=2\frac{\partial u}{\partial t}$$ could be in the form of a 1D heat equation

The Attempt at a Solution

$$\frac{\partial^{2} u}{\partial x^{2}}=$$http://www4e.wolframalpha.com/Calculate/MSP/MSP13061963dh6ehe94f57b000031ii70cfaaf938aa?MSPStoreType=image/gif&s=24

$$2\frac{\partial u}{\partial t}=$$http://www4e.wolframalpha.com/Calculate/MSP/MSP12211963dh8b7ca90e2d000034i7h8i9cfgdbif7?MSPStoreType=image/gif&s=27

I've tried to equate the two PDEs above to solve for $$\omega$$ and $$\beta$$ but I can't work out a solution for them, therefore I think I'm going about this problem the wrong way.

Any help would be appreciated, thank you.

 

[cepheid]

What if you ignored the given solution for a while and tried solving the PDE itself in order to obtain a general solution in terms of some constants? You could then compare that general solution to the given solution in order to find omega and beta by inspection. Is this possible?

 

[Hendrick]

Hi, do you mean try solving $$\frac{\partial^{2} u}{\partial x^{2}}=2\frac{\partial u}{\partial t}$$ without substituting u, first? Thanks

 

[Cyosis]

I haven't looked whether you differentiated correctly, but in their current form the relation between beta and omega is pretty obvious. Factor out beta^2 in your first expression and factor out omega in your second expression, equate and divide.

 

[Hendrick]

Hi Cyosis, I did make a mistake while equating. Thank you :)