Question

2. (Undetermined Coefficients... In Reverse) Find a second order linear equation L(y) = f(0) with constant coefficients whose general solution is: y=C et + Cell + tet (a) The solution contains three parts, so it must come from a nonhomogeneous equation. Using the two terms with undefined constant coefficients, find the characteristic equation for the homogeneous equation (h) Using the characteristic equation find the homogeneous differential equation. This should be the L(y) we're looking for. (c) Since we have used two terms from the general solution, let's turn our focus to the non- homogeneous part, tp. Find Wand and plug them into the homogeneous equation to find the f(t) we're looking for.

Answer 1

o since ypadest L[y]-fu) → The general solution is yget + lee6t+test Ipla estf 3te3F p" =zet+ 383t+gtest fp = bet +97e3t substitutup in the - 8 +8 Yar solution to homogeneas part Yp - Particular solution How th= c.p 2t +Cleht Llyj= f(t) in genesal * LC 3p] = fut) - Ila cient & qerzt 3 J"_By +12 Jp = fet) Whye ruin- Chaveste mistemola &#49783t)-8 (63t+37€) Here rioz. 182=6 +12 463t = fet) = (-2) d (86) are -> test+test-get-944et He factors of choreokristic ephel? tietest afet) therectestste equation is 1 -2e²t_3.test-fet) (8-2) (7-6) 20 182 88+12=0 1 fetta pel-ates! since the closceleristic opy © Hagesessel salu 82 88+12=0 The second oder deffraitid O como geneas differenteal reph (A2-8D +12)y co 0LEY) nerad =(02-8D+1234) Ley7= fet) fyll-pyltiny = -283t_34637) & Lry] = 9"-pyt124