Question

Find the Particular integral solution of the D.E (D^2 – 4D + 3)y = 20 cos x by the method of undetermined coefficients.

(a) yp = 4 cos⁡x – 3 sin⁡x

(b) yp = 2 sin⁡x – 4 cos⁡x

(c) yp =  – 3 cos⁡x + 4 sin⁡x

(d) yp = 2 cos⁡x – 4 sin⁡x

I got this question in quiz.

My question is based upon Method of Undetermined Coefficients topic in section Linear Differential Equations – Second and Higher Order of Engineering Mathematics

Answer 1

Correct option is (d) yp = 2 cos⁡x – 4 sin⁡x

For explanation I would say: ∅(x) = 20 cos x,we assume P.I in the form yp = a cos⁡x + b sin⁡x  ….(1)

and since A.E has m = 1,3 as roots,∓i are not  roots of A.E.we have to find a and b

such that yp” – 4yp‘ + 3yp = 20 cos⁡x……(2)

from (1) we get yp‘ =  – a sin⁡x + b cos⁡x, yp” =  – a cos⁡x – b sin⁡x, (2) becomes

 – a cos⁡x – b sin⁡x –  4( – a sin⁡x + b cos⁡x) + 3(a cos⁡x + b sin⁡x) = 20 cos x

(2a – b)cos x + (4a + 2b)sin x = 20 cos x –> 2a – b = 20 and 4a + 2b = 0, by solving we get

a = 2, b =  – 4 from (1) yp = 2 cos⁡x – 4 sin⁡x   is the particular integral solution.