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October 10th, 2012, 11:46 AM   #1
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(Q) Linear combinations of solutions

Hi! First post here, looking forward to browsing this forum

I've got a question. I just started studying Fourier analysis, and I've come upon a question that I couldn't figure out.

Q:
Suppose u1 and u2 are both solutions of the linear differential equation L(u) = f, where f != 0. Under what conditions is the linear combination c1u1 + c2u2 also a solution of this equation?

A:
c1 +c2 = 1.

Could someone explain why this is?
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October 10th, 2012, 12:10 PM   #2
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Re: (Q) Linear combinations of solutions

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) = c1f + c2f = (c1 + c2)f.

Clear enough?
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October 10th, 2012, 01:24 PM   #3
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Re: (Q) Linear combinations of solutions

Actually, yes!
I love pure algebraic explanations when they're simple enough.
Cheers!
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