Question

Solve the initial value problem: 4y" +12y + 17y= 0, y(1/2) = 1, y(7/2) = 1. Give your answer as y=... . Use x as the independent variable. Answer:

Answer 1

Given differential equation is - o 49''+12y'tity Consider the the corresponding auxilary equation is 4D² +12D +17=0 For axat batC=0 Roots -btsiyac are riara qa = D= -12 I JC127&_4017) (4) 2(4) Da -12 + 144 - 272 8 D = 12 I J128 i 8 D W -12 +852 8 D - 3 I sai

As the roots of complex are form 27 B there for solution is yer a ex CC,COS BX+C sinBx) Here edy - 3 x Yx) = e CC, cos(jam) +C& sin (rap) y'(x) 2x (-ac, sin(x) + S& Cecos (x) (с COS (823) + gasin (px)) By y (7/2) =1 a cos (IT) + Ca sin (SET By y'1 -58c, sin ( 51 ) + + sd ca Cos ( ) (1) CG, COS (STT) - eestinen put = 0 (just for solution simplishfication) ci 2 دیا [ ع

from ci C, COSO + Cg sino = liij) Co (-BENNO - cos@l+ cx (racoso - Sino) = D (6) Apply (+ sa sino +3.coso) x chip + (cos ®) xciv) Ca sa sindo +3 co sino cose +s& cocos po 2 COSO + COSO mlos sino COSO = Jasinot Ce Sa (sinde + corio) sa sino + Ecoso ce= sino + 512coso =

Using value of ca in (7) C Coso 552 1-sino (sinot coso) G coso =, 1- sinko 58 5 S2 sino caso G COSO = 52 sino coso 5. 4 Ci 552 sino 5692 COS coso 4 Hence Particular solution 's yo -1.5% e [(cos (190T- Suv sin (HU)) (cosa) +(91(99) + 5cc (90)) con]

If you want you can calculate constants c1 and c2 numerically by using scientific calculator and required decimal places.

I checked many times looking for error because such thpe of problem does not end like this but i i thi this may be.