Question

assume f(x)=a0+a1x+a2x^2+a3x^3+a4x^4...

4 Power Series Solutions to two important ODES 1. Find a power series (with two arbitrary constants) satisfying the following ODE S"(x) + f(2)=0 Write your answer in a closed form. 10 points

dont used this solution method pls because this not what the teacher want

Answer 1

00 "(z) + f(z) =0 Let us assume that f(z) = { an 2". = ao +212 +0x zºt- be the powen senies solution of the ODE nao moo, fles = { non zone and f(z) = nen-v) an zh-2 from 0, Mr & Mr 161369) maig 2" +Zanz" -o > { [(+2) Cnt) ana2 + an] ZP = 0 > (n+2)(n+1) An+2 +an zo' → An+2 = - an (n+2)(n+1) Let us look at the even coefficients Az = 2.00 -- azn = (-1)" (20) D!

Let us look at the odd coefficients as more a; z = To enhance Azman = (-)" content Hence the solution can be comitten as $(2) = 3 an zin -.f2)= zan - Ž - na . no Closed form of equation ② is, 1 f(z) = ao cossz.+ C Sinzl