1. The problem statement, all variables and given/known data 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. 2. Relevant equations Qualitative differential equations... no idea really what to put in this section. 3. 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.