- #1
royblaze
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Homework Statement
For the following diff. eqns (fcns of t)
X''m + λmXm=0
Xm (1)=0
X'm=0
X''n + λnXn=0
Xn (1)=0
X'n=0
Show that ∫XmXndt from 0 to 1 equals 0 for m≠n.
Homework Equations
Qualitative differential equations... no idea really what to put in this section.
The Attempt at a Solution
Using theory I am able to prove that the λ term must be positive in order to have non-trivial solutions. Using this, I am able to obtain explicit solutions for Xm and Xn respectively. However, when I attempt to take the integral I immediately am lost in how to show that their integrated product is 0.
My solutions are in this general form for both Xm and Xn, where C2 is some non-zero constant (to avoid trivial cases):
X = Cw*cos([((∏/2)+k∏)^2]x) for k = 0, 1, 2, ...
Any help would be greatly appreciated.