Question

(a) Given yı = et is a solution, find another linearly independent solution to the differential equation. ty" – (t + 1)y' + y = 0 (b) Use variation of parameters to find a particular solution to ty" – (t+1)y' +y=ť?,

Answer 1

They Selotion.com Given Gizet is a Solution of ty" (+ +09+yaº To find another Solution which is Linearl we use Method of reduction of order let y(t) = V(A) by, ( here (t) is tot - y(t) = V(H) et when @ be General Solution of 0 . . It It is the Differential is solution So solution it satisfies so equation & Differentrate (2 Y' (t) = V(H) et +V'CH) et = et (v (H)+v' (H)) Again differentiate y" (t) = et (V'CH) + 0"(t)) +et (u(t) + V'(I)) y" (#) = et (1") +2014) + () Potting values of y"(t) , y' (t) and y(t) in 9. y(t) is solution So it satisfies ODE ) Hlet (8°Cp) +91'CH) +2(3) - (++) (@4(0(4) +w\4) +VA) to -) et/tv"CA) ++14) +014)- 4164)-AV'(4) -)-') +2 = 0 et / tv" (t) + (2t -t -1) v' (H) {=0 tv") + (4-1) b'CH) = 0 Assume r' (t) = ult) V"(t) = U'(t) Butting in above - tut) + (4-1) u (A) which is Ist order Differential Equation.

IL Integra using vonable sebrable Solve t du = (-4) u = du = 1 t dt du = 114 - 1) dt the 6 of Integration - Intu(t) = lnt -t the =) Injucts = lnicts of - In ju(t) = -t = ult, - et u(t) = ct et since ult) = V(2) Putting we get & v'(H) = ctet - du = cite Integrating (do - a (tet at +62 = a 1 / 5 by Parts - V (H) = (fd et - se. et at} +62 = V(A) = CX-te-t-e-tę +(2 - at e-t 7 Ult=-e-ta (1+ d) + (2 Putting in o we get General solution as y (t) = V(A) y, (+) = |- éta (1+t) talet - 4 (t) = -((1+t) thet - which is General solution of

By General Solution - et and another solution which is independent of 6 is Yz (A) = -(t+1) Answer A Solution (6) By ③ General solution of 1 is y (t) = - G (t+1) +Get Consider the Non homogeneous Differential Equation ty" - (A+1Jy' + y = t2 - 4 Since complementary Solution (C.F.) of 4 is y (d) = -( (t+1) atlet I Let By method of reviation of Parameter its Porticular Solution (2) is

Let Yo (t) = -A(H) (4+1) +Bit) et 6 where Alt) and B(t) to be determine Y t = -A' (t) (4+1) - Alt +B (tet & Bitjet setting - A' (t) (4+1) + B' (t) et 0-0 Yé (t) = -Alt) + B (t) et Y"(t) = - A'(4) + B'CH) et +Bitlet (4) ] Soloròn c# # - AY"(t) - 4+1) J' (t) + Yold) = t² -td-A'CH) +B'ch et +B48)eff-(++){-A [+) +Bet/ + ? - Alt) (4+1) + B (H) ets=t? - - t A'CH) + det 8' (t) = 1² =) - A'(d) + et 8'(t) = d ② from 6 and 6 6 - A'CH) +et B' (H) - LA CH) (4+1) + B (teth=t - - A'(+) { 1 + t +1h = t. - A' (t) = - t = -1 4+2 1225 A Al(t) = - 4 1 - 2 1 4 2 1 = 28 24 2 - 1 H4 = 2 ln (4+2) - t Integrating

And B' (t) = (42+ + Jet Integrating site t) at (Due to time Cons ula Please solve BIH) = 16 Potting Alt) and Bit) in 6 Yy14) = -J2 An (442) –{(441) 46 Le Hyde) (4) = 4(44) -2 4+1 den (42)7 et like Yok This is required Particular Solution Answer