Question

Problem 13. (1 point) Consider the differential equation y" + ay' + By=t+e3t. Suppose the form of the particular solution to this differential equation as prescribed by the method of undetermined coefficients is yp(t) = Alt + Apt + Bote3t Determine the constants a and B. a= help (numbers) B= help (numbers) Note: You can earn partial credit on this problem.

Answer 1

sol. et y" + dy't By- tt e3t o 1 rest Here Yp H= Ait²t Art + Bo test 7 then yo = 2A1t+ Ao + Bo e3+ + 3 Botest Y" = 2A, + 3 B. e3+ + 3 B. e3+ + 9 Bote 34 using ☺ and @ we have 2A1 + 6 Boe3+ +966 +e3+ + 2 Alex+ + Aod + Bod e$* +3 fo otte 3+ + AiBft Aoß t + ß Bo te34 - t+e34 (2 A1+ Aox) + +(& Aia tAoß ) + t ² ( A,B) + e3t 16 Bot Box) + test (9 Bot 3 Box + P t Box) + test (9 Bot 3 Box + ß Bol=ttest | 9 Ait Ao L=0 = 9 AIR + Ao ß- Alß = 0 9 Bo + 3 BOX + B Bo = 0 6 Bo + B.2=1 Ao X = -2 A1 2A1X + Aoß = 1 AIB = 0 38. a + BB. = -9 Bo 2B0 = 1-6Bo 5-9A) Ao TT 9 Al Y - Os -qBo 3 Be 1-bbo X I Consider Augmented matrix : - 9 Al : 2 Al O 3 Bo Ao AI Bo -q Bo - os ? ? 1-6 Bo J р Во Во

Sing we have AIB=0 Also oo B=0 rao o If Alco then -renom O-O TI F-6BO E å om í 1-4 à eco 1-6BO deo TI Hox=0 AOB = Be to 3a +B=-9 - so DE is y"-9y: tt és Then y= Alt²t Ast + Bote 3t isi p.s. with Alco If Alto then [B=0 = 3B0 2 + Bo B = -980 3BoQ = -9 Bo * Dife is y"-3y' = t test but then Yo= Aitt hot + Bo test is a solution of Die so final answer = x= -3 Bao