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M AA WIKITICAL ASSOCIATION OF AMERICA < wetwork matt263_fa00197514 5: Problem 4 Previous Problem Problem List Nest Problem (1 poll) The solution of the differential equation x + 2y = 80.com/8) - 90 sin(Sa) subject to the witial conditions (0) and (0) - 9 Preview My Answers Submit Answers You have attempted this proble m es You have unlimited empts remaining Emall We Work TA

Answer 1

Solan:- Given problem, " +257 = 80 cas 52 - 90 SinEL – 10 subject to (0) = 0 and y'(o) = g. - - The auxiliary equation of ① is mrt 25 = 0 ma +52 Dr+25 . Dr725 . o 6425 The complementary function of eqr 6 Is = G cosx + casinex Now P. I = I (800085-gosinux) $426760552% 30 sin52% Now 125 COSEX Real porrt of 425 eisx peal port of 2152 ps3 y 25/0} - Real poort of 15x or - 2010 til Real port of else oid it on ki - Real poort of is (1 + 8 ) til Real perrt of pise 1 (- for + b ) { Dr1010 10tD

1001 - Real pornt of six I (1) - Real pornt of & six = Real part of (cos5x tisinx) xSin 52 Also, t2 sinux - Img point of 25 eu = Ing. part of six (6+51) 725 Innge part of all I be tole Tony. pornt im 3 3 Img porrt of sie bei (1-10 = Tang.port of esiko - Ing. pornt of six x IODI 107 - xcos su - To

680x sin5R goxcos5x P-I = 10 = 8x sinux + 9xc0B5X. . The general solution of the equation o IS & & (x) = C-F + P. I → Y (2) 920052 +G SI5X + 8x Sin5z + gx cas 5 - - for Given, & (0) = 0 7 G=0 Giren 7 (0) 29 Now, 7(x) = -54 sin5x + 56 COB 5X + 8 sinux +402EOS 5X + 92035x 195xsinex 7 10) = 0 + 562 +9 7 9 = 56+ 9 > z= . The solution of the given equation is (x) = bestant + 8x sin5x + 94c055x