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Homework Statement
The set of solutions of an ODE of the form y''+by'+cy=0 forms a vector space. To convince yourself of that, prove that axioms 1,4,5 and 9 of the definition of a vector space hold for this set of solutions. (You may want to check the others as well, but no need to present their proof.)
Homework Equations
The Attempt at a Solution
I started to prove the first axiom as follows:
Axiom #1: If u and v are objects in V, then u + v is in V.
If we let c1u and c2v be two solutions to our second order differential equation, then c1u + c2v must be a solution for the second order differential equation for any constants c1 and c2.
F(y) = 0 = (c1u+c2v)’’ + b(c1u+c2v)’ + c(c1u+c2v)
0 = c1u’’+c2v’’ + bc1u’+ bc2v’ + cc1u+cc2v
0 = c1u’’ + bc1u’ + cc1u + c2v’’ + bc2v’ + cc2v
0 = c1(u’’ + bu’ + cu) + c2(v’’ + bv’ + cv)
0 = c1F(u) + c2F(v)
0 = c1(0) + c2(0) = 0 + 0 = 0