Question

Find the first five nonzero terms in the solution of the given initial value problem. y" + xy' + 2y = 0, y(0) = 5, y' (0) = 3

Answer 1

Here assuming y as a series in x, with the help of given inial values and differential equation; we get the desired solution.

Let y = annn be the series solution, n=0 of the given differential equation. differentiating. y'= E nannni then on word ni n=1 Now again differentiating worot ni y" = [(n-1) a n(n-1) anna-2 n=2

Criven initial value 46.) = 5 and y'(o)=3 Now y(0) = 5 = got Gn.o. n=1 9=5 and y' () = 3 = Enan.o + 19, na2 9. = 3 Thus we have 90=5, Now putting these values in the given. differential equation

Z n(n-1) an nn-2 tn nannh- n=2 n=1 + 2 2 anan anan = 0 no I n(n-1) an nh-2 + nannh h=2 h + 2 annno hao Now we can write by replacing night2; Z n(n-1) annnn, g(n+2) (n+) antza h%3D0 ne2

Abus 5 (142) (11) 2 1) + รีก 40 hon1 + 10 hะ *C 0 = 44 % ดsu 9 2.): + 2(142)(+) 90+1 72 + = ร.ศ. ๒-๓ +124 17 + 2 .40 = 0 > 2 (92 +4.) + (h+2) (n+1) 5+) ) ใค+2 + hs | + 17 : Now comparing the coeff on both rides: - 92 - - s = -5 2 (9) 196) = 0 (n) (h+1) 91+2 +(12) 59 and रि५

Put n=1 +2) 9,=0 (1+2) (1+az + (1+2) 693 +3. (3)=0 (:9=3) az = -3/2 Put n=2 (2+2)(2+1) 94 +(2+2) 9,50 (-5)=0 12 94 + 4 (: 4,5-5) => 94 = 5/3 Now, the solution. y = { anan h= 90+9, n+an²+ 9 ₂ 73 +9 404 + 2 anno This first five non-zero 8: 5 + 371 + (-5)m? + (-3/2) 13 + 5/04 n = 5 terme of