Question

Find a particular solution to the differential equation using the Method of Undetermined Coefficients. y" - y' + 484y = 22 sin (22) A solution is yp(t)=0

Answer 1

D We know that when the differential equation is in the forma de + Plu) over +91) y = Rim) the complete solution will be taken as Y = e.Ft YP where eof is the complementary function and Yp is the particular solution. "To find Yp, when Rin) Containts siman or cosan ay If sinan or Cosax is not in the elf, them take Yo A simandB Cosan. fake Yp ma by If simon or cosan is in the cof, then x (Asinanny B cos am) Given differential equation is Y"_4'44844 = 22 sim (224) Y = A e (Asto) be a trial solution of the differential equation y" 4'44844 20 : The auxitiary equation is m2-m +48490 1 IV 1-4.484 21 E I I 1 - 1936 ret, 1 1995 2 1931 2 215 The cof is setela Cos (3/2154)+cz 8in (BJ 215 ) wobitouny anstante where A C₂ are

D402 403 AS, Sin (224) is not in the cor. So, we take parficular & Solution is of the given differential equation is in the form Yp = 'A'Cos(224) + B Sim (227) -—(11) Then, Dyp = -224 simpat) 4:22 B Cos (224) Dyp Dyps - 223A-COS(22+) – 22?xB sin 227. we know that parficular solution satisfy the differential equation . so, D? Yp - DYp + 48448 = 22 Singet. >> 62298 Cus224 – 20%B59m 29+) - (-22 A SM24221) +298 Cos (024)) + 4841 Alos 217 +BJfn227) © 22 3in (224) (-4841 484 A À 22B +4846) Cos 224 (-4848.722 A +4848) 3in22+ = 22 Sinaat 51 - 22 B Cus (224) +22A Sîn(22+) = 22 Sin (224) How equating the coefficient of sm 624) and Cos:1924) we get 22 A = 22 ay A=1 -22B 20 57 Bao i Then the parficular solution is Yplt) = Cos(224) [using (1) and