), which you solve using the given boundary conditions (like ) to find specific values for and their corresponding eigenfunctions .
Since the PDE is linear, any linear combination of your product solutions is also a solution. Express the general solution as an infinite sum :
. This often involves calculating a Fourier Sine or Cosine Series for the function using orthogonality integrals . For a sine series on , the formula is:
An=2L∫0Lf(x)sin(nπxL)dxcap A sub n equals the fraction with numerator 2 and denominator cap L end-fraction integral from 0 to cap L of f of x sine open paren the fraction with numerator n pi x and denominator cap L end-fraction close paren d x