Question

1. [10pts] (a) Solve for 2 > 0 the DE: 20%Y" + 102’y" +59xy' + 45y = 0. Answer: Solution: (b) Solve the DE in Part (a) for r <0.

Answer 1

( 1 ) Yedico the (Oy x ²4" - 104 " +59xy'+ 45y = 0 linear romogeneous differential equation on into line arr differential equation with equation with contant coefficienta, we pu put el X. OY, 2= good and @ Now, dy da dy dy al dz d2 da y'= R. dz where =Oy → xy' = dy dz d dz dy 2 dz = 1 2 a da yo che non tenet I dzy dz x dar da dzy - use d (d 2 - 1) y (-) yu = o(0-1)y Similarly, xy" = 0 (0-1) (0-2) Substituting these values of from a into the equn 010-1) (0-29% +100(0-1)y +59 844 454=0. +{(03-387+20) +(000-100)+590 +453y = 0. 29 we get

7 (03+70°+519 +45)y=0 >> { DM9 +1) + 69 (0+1) +45(0+1)y = - }(0+1)(0760+45)}y = 0 Let y some be the trial solution of than the auxiliary equation is (m+1) (m+ 6m+45) = 0 -) mot, The solution of is y = ge ? + 0-32 (GC0067 + Cylin 62) 7 where G, C, are -3 +61 three arbitrary contants Putting the value of z in we get the required solution of the differential equation for ryo is Я. {cy.cop (6 Anta) +G8 in (6180) 19.00B(Antes) Cg sin( Inx"))} for ayo. + x a ole ole + -

( 6 ) can The solution for a co be obtained from that of 270 by replacing 'n by L2 overyeruere. Thus, the solution of the differential equn ♡ is y = Caloocan (26) 213 + Cg sincan Cab ole for x.co. [ By replacing in 8 ige by -