Question

You have to use Method of Undetermined Coefficients

LaTech Zill8e Ch4Sec4: Problem 3 Previous Problem Problem List Next Problem (1 point) Find the solution of y" + 5y' = 200 sin(5t) + 300 cos(5t) with y(0) = 8 and y' (0) = 4. y=

Answer 1

Given, y" + 5y = 200 sin (st) + 300 cos (5+) yo) = 8, yo = 4 with Il Auxillary equation is m?+ 5m = 0 - m (m + 5) = 0 L i mz0, -5 . . 1: complementary function Yatie te _ _ : - 5 + =) Y= ct czë. Now, have to guess particular solution we bet Yoo A sin (5 t) + B cos (5) .: y.' - 5 ACOS (5t) – 5 B sin (5+) " --25 A sin (5+) - 25 B cos (5t) in Now, substitute values of Up, Up & you given_differential equation - 25 A sin (5+) – 25 B.cos (5t), + 5 5A cos (5+) – 5 B sin (5+)] = 200 sin (5t) + 300 cos (5)

- 25 A sin (St) - 25 B Cos (5t) - + 25 A Cos ( 5t) - 25 B sin (5+) ___ = 200 sin (st) + 300 cos (5t) :: (-25A - 253) sin (5t) + (254-253) Cow (st) = 200 sin (5t) + 300 cos (5t) compairing coefficientstrive have - 25 A - 25 3 = 200. 25 A 25B = 300 Adding these two equations, - 50B = 500 - -B 500 to- -50___ and 25 A - 25B = 300 - ___25A - 25(10) = 300 : 25A + 250 = 300 25 A = 300 - 250 .25 A = 50 Az-SO-2 25 ... Y p = A sin (5t) + B Cos (5) y = 2 sin (5€) -10. Cos (5+).

2. General solution - y = + Yp = 6 + (zēst + 2 sin (5+) -10 COS(SE) Now, we have y(0) = 8, y (o) = 4 .: 4 (0) = (y + lą ė + 2 sin (0) – 10 Cos (0) 8 = C, +62 -10 +CZ = 18 - Alex - 0 y'(o)=4 |:. y' = – 5 Co ē5€ + 10 cos (5t) + 50 sin (St) - y'LO) = -5 C, e +10 Cos (0) + 50 sinco) 4 = -56+ 10 -6= -562 I! From @ 40 . c=18- C2 = 18 - 14 :. Solution is, 8 y =- , 6 -5t . . . tē ērta sin t5t) =To cost5t)