Given the expression: $u(x,t) =e^{-Ct} y(x,t) - v(t)$
How would one approach this when looking for the Fourier series solution, or where to start?
Is it possible to see from this which Fourier coefficients would equal zero allready?
With the following conditions:
$C > 0 ,\qquad 0 < t, \qquad 0 \le x \le 1$
$y(0,t) = y(1,t) = 0 = u(0,t) = u(1,t)$
$y(0,0) = y(1,0) = 0 = v(0) = v(1)$
Don't know if this is of importance:
$v = v(x)$ satisfies: $v''-Cv=-f(x)$
$u = u(x,t)$ satisfies: $u_{xx} = u_{tt}-Cu= f(x)$