Set up but do not solve for the appropriate particular solution yp for the differential equationy′′+9y=-3xsin (3x) using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x). In your answer, give undetermined coefficients as A, B, etc.

Question

Kellen Solomon

Mathematics

31 August, 08:47

Set up but do not solve for the appropriate particular solution yp for the differential equationy′′+9y=-3xsin (3x) using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x). In your answer, give undetermined coefficients as A, B, etc.

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Pierce Sloan · Answer 1

First find solution to homogeneous DE y" + 4y = 0:

Characteristic eqn = > r^2 + 4 = 0 = > r = + / - 2i

y_h = C_1*sin (2x) + C_2*cos (2x)

So our particular solution will be of the form yp = (Ax + B) * sin (2x) + (Cx + D) * cos (2x)

However since Bsin (2x) and Dcos (2x) are already solutions to the homogeneous DE, we multiply the yp through by x:

yp = (Ax^2 + Bx) * sin (2x) + (Cx^2 + Dx) * cos (2x)