Question

ters) SUIVe y + y - Sez. Find the general solution to ty" + 3ty' +y = 0 given that yı = t-1 is one solution. 1x , ,- 8+ (0) - 1 10 _1

Answer 1

By method of reduction of order
t'y"+3t y'+y = 0 = is one selution given Let the other Solution be for Sore fundtion w t) Y2 = w (t) Y, 9= Wit) y = twit) - wit) %3D and (w'(t) +t w" (t) – w'f(2)) t"-2+ (t w't) –w (e)) %3D P"t) – 21"w't) + 2t wlt) M. Y,"= t'w"(t)– 2tw'(t) + 2 w (#) Substituting thon in the given diffrential eguation tu"t)- 2t w'(#) +2 w (t))+3t ( tw'(#)-w (t)+ wt) tw"(t) - 2 w'(+) + W(t) + 3 w'(t) - W(t)+ W(t) t w"(t)+ w'(t) = 0 > w" (t) = v'(t) = dv w'(t) = V(t) Let %3D 4P. Substituting Hak in equation O t dv + V(t) = 0 By sepusakion of Vasiabler AP

Antigeating On both Sides (natural logarithm) In (v) = -Ln (t) +Incci) untigeation Comstant Luhele Im (c,) is an vt = C, ('2) = (1M)4| V= w' (6) %=D 부 du t. 4P that is thanfate wit) = Jw'c)dt Wt) =| (where log) is Same as natuial logeadithm In()) wit) = C, log(t) + C2 de %3D intigeation Gontant wit) = c, log(t)+Cz Theufhile the othch solution is y,4) = (, log(t) + C3 )" ,t) = Ci log (t)+ c2 C2 is an to The general Solution is y4) = A,Y, () + B,Y, {#) for Some albitsaly Constants A, B, y#) = A+ + R ( la t)+) YA) = A, + B, C2) + B, C, log4) (replace A,tB, Gg=A 4(#)= At+B log (t) † ' | fr Some aubitoasy Constanta A, B f B, GG = B )