Question

9. For the differential equation: ?′′ + 2?′ + 5? = ?(?)

(a) Write down the auxiliary polynomial associated to this differential equation and find the roots of the auxiliary polynomial.

(b) Find a basis for the real solution set of the associated homogeneous differential equation ?′′ + 2?′ + 5? = 0.

(c) For each ?(?) that follows, write down the trial particular solution to ?′′ + 2?′ + 5? = ?(?), with undetermined coefficients. Do NOT solve the constants that arise in your trial solution.

(i) ?(?) = 120?^−?

(ii) ??(??) = 68 cos(2?)

(iii) ??(??) = 3?

(iv) ??(??) = 75?^−? sin(2?)

(d) Find a particular solution to ?′′ + 2?′ + 5? = 100 cos(2?) (show your work)

Answer 1