Eigenfunction expansion

Question

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is $$\phi=A\cos x\sqrt{\lambda}+B\cos x\sqrt{\lambda}$$ but how should I decide A and B? Is it by system$$\phi'(0)=\alpha, \phi'(1)=\beta$$ or it should be $$\phi'(0)=0, \phi'(1)=0$$ and why? After getting eigenvalue and eigenfunctions, what should I do? I hope somebody can give me a answer with details. Thanks.

Answer 1

Your characteristic equation should be $\phi'' - \lambda \phi = 0$

The roots are ths $\pm \sqrt \lambda$ , and $\phi = A e ^{ \sqrt \lambda x} + B e^{- \sqrt \lambda x}$. This can be rewritten in terms of cos and sin, although if $\lambda$ is complex you will still need some exponentials. Learn more here: http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf

At this point you really want u'' = u = f(x), so take $\lambda = \pm 1$. Then u'' = u = f(x) = $Ae^x + Be^{-x}$.

Now that you know what u is you can compute u'(0) and u'(1), and find A and B from the initial conditions.