Question

By using the method of undetermined coefficients, find the general solution of the following differential equation

(f) /' + 4y = cos 2x.

Answer 1

?) '' +4a = Cos 21 1. let y= einx 2 m2 + 4 = 0 - м = +25 №3 пе () = е свъ (2) +C25in (2х).

Noco yp in of the form Ax Cos (2x) + Bx sin (2x) So, yp = Ax cos(2x) + B x sin(2x) yo'= A cos (28) – 24x sin (2x) + Bain (2x) + 2 Bx cos (22) ypll - 2A Sin(2x) - 2A sin (22)- 4 Ax cos(2x) + 2B Cos (22) 1 +2B COB (2x) – 4 Bx Sin (26) = - 4A sinex)- 4Bx sin (2x) + 4 B COS (22)-4Axlos (22) So, yp" + Up = 'los (2x), => (48-4Ax) eos (22) - (4A+487) sin(27) + 4 Ax Cos (22) . + 4 Br sin(23) = cos(2x) => 4B cos (2x)- 4A sin (2x) = Cos (27)

comparing both sides we have 48=i =B = . and ' 4A =0 » Azo There fone Mp = 0.x cos (2x) + sin(2x) = Lin (2x) Therefore the general solution is a y = a cos (2x) + C2 sin (2x) + 2 sin (24) = ei cos (22) +(62+ %) sin(2x)