Transform the given 2nd order initial value problem into an initial value problem of two 1st order equations?

Question

Transform the given 2nd order initial value problem into an initial value problem of
two 1st order equations (by letting x1 = u and x2 = u'), and write it in matrix form:
x'= Ax + b; x(t0) = x0. (No need to solve!)
(a) u'' + 0.25u' + 4u = 2 cos 3t; u(0) = 1; u'(0) = 2
(b) tu'' + u' + tu = 0; u(1) = 1; u'(1) = 0 Transform the given 2nd order initial value problem into an initial value problem of
two 1st order equations (by letting x1 = u and x2 = u'), and write it in matrix form:
x'= Ax + b; x(t0) = x0. (No need to solve!)
(a) u'' + 0.25u' + 4u = 2 cos 3t; u(0) = 1; u'(0) = 2
(b) tu'' + u' + tu = 0; u(1) = 1; u'(1) = 0

Explanation / Answer

(a) Let u' = v. Then, the 2nd order DE transforms to v' + 0.25v + 4u = 2 cos(3t). ==> v' = -4u + -0.25v + 2 cos(3t). By setting x = (u, v)^T, we have x' = Ax + b, where A equals [0 1] [-4 -0.25], and b = (0, 2 cos(3t))^T. ----------------- (b) Let u' = v. Then, the 2nd order DE transforms to tv' + v + tu = 0. ==> v' = -u + (-1/t)v. By setting x = (u, v)^T, we have x' = Ax + b, where A equals [0 1] [-1 -1/t], and b = (0, 0)^T. -----------------

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