How to find the eigenfunctions of a differential operator.

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To find its eigenfunction $f$, it is equivalent to solve $Lf=\lambda f$, that is, $$\frac{d^2f}{dx^2}=\lambda f.$$ This is an second order ODE with constant coefficient, which can be solved. After finding all the possible solutions for $f$, we can consider the normalized condition and initial conditions to find the specify $f$.

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Updated on August 01, 2022

Comments

  • ODP
    ODP 3 months

    Consider a linear differential operator $$L=\frac{d^2}{dx^2}.$$ How would one determine that the normalised eigenfunctions of $L$ are $$\phi_n(x)=\sqrt{2}\sin{(n\pi x)}?$$

    • tired
      tired over 6 years
      which initial conditions do u have in mind?
    • ODP
      ODP over 6 years
      I think y(1)=Cy(0) but I will double-check and get back to you later.
  • ODP
    ODP over 6 years
    OK cool, so with boundary conditions $f(0)=0$ and $f(\frac{\pi}{4})=1$ I obtain $f=A\cos{\lambda x}+B\sin{\lambda x}=\sqrt{2}\sin{\lambda x}$ where $A=0$ and $B=\sqrt{2}$ from the boundary conditions. But then how does one determine that $\lambda_n=n\pi$?