Question on orthogonal eigenfunctions

[mt8891]

in this book I have by G.L Squires. One of the questions is:

if \phi₁ and \phi₂ are normalized eigenfunctions corresponding to the same eigenvalue. If:
\int\phi₁*\phi₂ d\tau = d

where d is real, find normalized linear combinations of \phi₁ and
\phi ₂ that are orthogonal to a) \phi ₁ b) \phi₁ + \phi₂the part I'm having trouble with is finding a linear combination. The book gives: c1phi1 + c2phi2 (to shorten it) as the linear combo but yeah, what's the deal with that. they give a solution for a but yeah what is going on there. what's the process involved. yes, I am basically asking for a "tutor" so to speak, so I am n0t asking for any old person to help me but someone who is willing to explain it without becoming angry that I know not of such things.
thanks in advance!

 

[mathman]

Let L be the linear combination (depends on c1 and c2). In both cases you have one (of two) equation L.L=1.
(. means dot product, which is the integral you described).

For a) the other equation is L.φ₁=0.
For b) the other equation is L.(φ₁+φ₂)=0

You wil have in both cases two equations in two unknowns (c1 and c2).