Question

Consider the differential equation L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and...

  1. Consider the differential equation

    L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and suppose L[yf] = f(t) and L[yg] = g(t).

    1. Explain why yp = yf + yg is a solution to L[y] = f + g.

    2. Suppose y and y ̃ are both solutions to L[y] = f + g, and suppose

{y1, y2} is a fundamental set of solutions to the homogeneous equation L[y] = 0. Explain why

y = C1y1 + C2y2 + yf + yg is the general solution to L[y] = f + g.

(c)  Use this to solve the initial value problem y′′+4y=t2+3et, y(0)=0, y′(0)=2.

  1. Use variation of parameters to find a general solution to the following differential equations.

    (a) (2pts)y′′+y=tant, 0<t<π/2. (b) (2pts)y′′−2y′+y=et/(1+t2).

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