Donation of arbitrary constant

Question

s=c1e^-3t + c2e^4t cos 5t + c3 e^4t sin 5t + 100/123

a. sttt + 5 stt + 17 st - 123 s = 100

b. sttt + 17 stt + 5 st - 100 s = 123

c. sttt - 15 stt + 17 st + 123 s = 100

d. sttt + 5 stt + 17 st - 100 s = 123

Answer 1

s is goverened by the exponents. -3, 4 +- 5i along with a constant shift.

The first three exponents are annihilated by the differential operators

(d+3) (d - 4+5i) (d-4-5i). which arranges to d^3 -5 d^2 +17d +123.

Applying this to s yields its application to the constant term

(123)* (100/123) =100

So

sttt - 5 stt + 17 st + 123 s = 100