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

Mathematics

Krish Dougherty

4 May, 05:49

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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Dillon Whitehead · 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)