Find a formula involving integrals?

Question

Find a formula involving integrals for a particular solution of y'''-y"+y'-y=g(t).

Answer: Y(t)=(1/2)∫(from t_o to t) [e^(t-s)-sin(t-s)-cos(t-s)]g(s)ds

Answer 1

Hello Sun,

The problem you pose is probably best handled by the method of Variation of Parameters. We first seek to solve the associate homogeneous equation

y''' - y'' + y' - y = 0

which is solved by finding the roots to the characteristic equation

r³ - r² + r - 1 = (r - 1)(r² + 1) = 0

which lead to the solution of the homogeneous equation

y = c₁y₁(t) + c₂y₂(t) + c₃y₃(t) = c₁e^t + c₂cos(t) + c₃sin(t).

The Wronskian, W(t) = W(y₁,y₂,y₃)(t), turns out to be

W(t) = 2e^t,

and, defining the determinant W_k to be W with the k^th column replaced by (0,0,...,0,1), you will find that the solution (via Cramer's Rule and one integration) looks like

Y(t) = ∑_k=1³( y_k(t) ∫_t0^t ( (g(s)W_k(s)) / W(s) ) ds.

From this formula (which is essentially the result of the Method of Variation of Parameters), and computing the W_k's on your own, along with W, I think you should be able to reproduce the answer given.  If not, let us know where you are getting hung up, and what you've tried so far.

Best of luck, and I hope this helps!

Regards,

Hassan H.