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- Homework Statement
- Verify the PDE has the following solution
- Relevant Equations
- ##u(t,x)=f(x-t)+g(x+t)##
I suspect this problem is about the wave equation ##\partial_t^2-\partial_x^2=0## commonly encountered in physics. I tried a search for information but I could not find help.
Attempt at arriving at solution:
So I took the partial derivatives of ##u(t,x)=f(x-t)+g(x+t)##
##\partial_t u=-\partial_tf(x-t)+\partial_tg(x+t)##
##\partial_t^2 u=\partial_t^2f(x-t)+\partial_t^2g(x+t)##
##\partial_x u=\partial_xf(x-t)+\partial_xg(x+t)##
##\partial_x^2 u=\partial_x^2f(x-t)+\partial_x^2g(x+t)##
Giving
##\partial_t^2-\partial_x^2 =0##
##\partial_t^2f(x-t)+\partial_t^2g(x+t) -\partial_x^2f(x-t)-\partial_x^2g(x+t)=0##
##\partial_t^2f(x-t) -\partial_x^2f(x-t)+\partial_t^2g(x+t)-\partial_x^2g(x+t)=0##
I try to reason if the last line is true, then so is ##\partial_t^2-\partial_x^2=0## because it shows
##\partial_t^2f(x-t) -\partial_x^2f(x-t)=0, \partial_t^2g(x+t)-\partial_x^2g(x+t)=0##.
Although it does not feel sufficient.
For ##u=f+g## I can set ##t=0## in ##u(t,x)=f(x-t)+g(x+t)## and just get back ##u=f+g##.
For ##\partial_t u=-f+g## I can take ##\partial_t u=-\partial_tf(x-t)+\partial_tg(x+t)## and set ##t=0## to get back ##\partial_t u=-\partial_tf+\partial_tg##. So the last condition is not satisfied unless setting t=0 somehow means removing ##\partial_t## from both terms.