Show that the solution of the initial value problem L[y] = y__ + p(t) y_ + q(t) y =

Chapter 3, Problem 17

(choose chapter or problem)

Show that the solution of the initial value problem L[y] = y__ + p(t) y_ + q(t) y = g(t), y(t0) = y0, y_(t0) = y_ 0 (32) can be written as y = u(t) + v(t), where u and v are solutions of the two initial value problems L[u] = 0, u(t0) = y0, u_(t0) = y_ 0, (33) L[v] = g(t), v(t0) = 0, v_(t0) = 0, (34) respectively. In other words, the nonhomogeneities in the differential equation and in the initial conditions can be dealt with separately. Observe that u is easy to find if a fundamental set of solutions of L[u] = 0 is known. And, as shown in 16, the function v is given by equation (30).