Subject: Differential Equations

solve the following questions ( handwritten solution only neatly and clean box final answer) thankyou!

answer 4d and 5

See correction below

vector space of all functions 4 . a) Let I be an open interval. For any ye D, CT) = atleast twice diffentiable on I, (apts ) define Lly) = any" + a, y + ao's, Mcy)= bay" + by'+boy, ['=dax] where do, bo, al, by, a2, by are continuous on I . Prove that if Lly) = Mcy) for all ye D, (I) then, ap= bo, 9, = b, and 9,=by on I vector space of all b) Suppone that for le=0, 1, 2, aRE Dye (I ) where Do (I) = continuous function on ]) (3pts )' Prove that L(y ) can be written in the form. ( Py' +Qy ), where P, QED, ( I) if and only if ( a, - a ; )' = a. .C ) Let Lays = y" - (20 +5 )y' + (40+ 6)y (4.0. 1 ) ( 1pt ) is show that if ex is a solution of ( 4. 0. 1 ) then d= 2 . (5pts) iis Consider the IVP : L( y) = 0 , 4(0) = 1, y'co) = 3. let Gax) = ) etthat . write the solotion of the 1UP, in terms of G(x). d ) het Mc y ) = 4"+ ( 2 x + 5 )y' + (4x+8)y . . Suppose that for some ue D, ( 6-00, 00) ) and P, G E D. (6-0, 001), UMcy) = (By'+ Qy ). ( apts ) ;) Give a and order differential equation to which u is a solotion . and give a solution to the obtained equation . ( 5pts) i ) solve : M( y ) = e 2x 8x if x e (-oo, 0] (Kepts ) 5. Let poo = Linecoax if XE (0,00 ) solve : y"-24' -34 = Pag y co) = 0 , y(17/2 ) = 12 e 35/2Math 122 Problem set corrections : 1. c . ( 42 - x tedx + (8y ? - x + 2 x4 + 1 ) dy = 0 4 . di using 4. 6. ) give a second order ... 3x 5 . 12 e P ( x) =