Question

find the general solution
d2θ/dt2=(1/c)d2θ/dx2
should be in the form
θ(x,t)=(Acos(ωx/c)+Bcos(ωx/c))(Ccosωt+Bsinωt)

Answer 1

nice partial differential equation, the substitution takes forever, but we can try

Answer 2

set it up the same as solving the heat or wave equation letting it = \[-\omega^{2}\] then solve and run through your 3 cases, repeated roots, distinct, imaginary

Answer 3

might have to do fourier which is brutal let me know if there are any short cuts

Answer 4

I already answered this but appears to have been deleted.

To restate the usual sin cos solution in your desired form is simply a matter of some rather tedious algebra.

Answer 5

im ok with algebra and solving differential equations however i havent encountered a differential equation of this type before. It would be a great help if somebody could just run me through the first few steps. thanks
and sorry it should be \[1/c^2\]

Answer 6

oh this is a 10 mark part of a question question from a previous years exam, so it should take about 20mins tops. it is for free torsional vibration of a shaft, first part of the question is to ascertain the diffential equation and second part is what i have posted.

Answer 7

Your pde (with 1/c^2) has a similar form to the wave equation, what c happens to be depends on the particular type of problem being examined.

If u sub sin kx cos kct (this on general principles) (do the partials) that should satisfy the pde for some constant k and so it is a solution (which u can recast in the form required by your question).

One would not usually find a general solution, but particular solutions based on boundary and initial conditions.

This might be helpful

http://en.wikipedia.org/wiki/Wave_equation#General_solution