Question

Solve the ODE/IVP: 4x^2y'' + 8xy' +y=0, y(1)=2, y'(1)=0

Please help me solve this using series. Thanks

Answer 1

ODE Ans: Given 4 x²y"+8xy't j = 0 y(t) = 2 y60) 20 Let y Ž Commare mco mor-1 (mt) Cmn yl M1 y" E (mto) (m+ M-1) Cm. mtMa2 M-2 Substituting the values in above equo mtr-2 4 x2 E (m tre ) (mtrol) cm x + M22 mtr on Ž (mar) Comise min-1 + E Cms mo mtr MAR at & ym + M) (M+M-1) (ma + £ 8(mare) Com xm M=2 MO! t Ecman mar 0 n M-2 E 4 (mtr) (MAM-1) Cmn + ² 8. (mari) cm a teema cmum M20 May

2 .: se to then ay 84&mtr) (mtm-1) (mint Ě 8(mtr). Comant (mano M20 MO Now equating the coefficient of x with a (enca-1) +84 +1) Co=0 =7 (442-44+8r+.).C020 ay (4m² + yr +1) Co 4 M² +40+1.co 7. (r + y) = 0 -Y2 -12 the roots are same. 20 Here Now equating the coefficients of us with Zero, 48 +2) CS+-1) + 8 (st) stes 20 => 4 (str) { Stu-l tu žest Cs {4(544) (Sta+3) +1}es 7 Cs Hence, y = n { Coto Mt Can? t ......} Enco (*6=622.... "; = M2=/2 then 2 Here yi -12 Im="12 X

And Y2=lna (",) alna -V20 Hence, Y(n) ci git (2 42 -Y2 - cia -72 + C2 a 2 2 + Crna rn - ya) a von (eiteainn) Elnaro Griven that. yo) -> C Again y'com) (citczlinx) + 4.62 y (1) => 29 te=0 -> Hence, y (n) = 2 / 2 (2+ lax) re a C22 1.