Question

3. First, here is a summary of the method of variation of parameters (Braun 2.4). Given a general linear second order ODE of the form with p, q and g continuous on some interval I that contains the initial condition, and given that you have a fundamental solution set gi(t) and y2(t) to the homogeneous problem Ly]-0, one can find a particular solution as follows. [Follow along on pg. 154 of Braun] . Let yp(t) = ui (t)n(t) + u2(t)m(t), where ui and U2 are unknown functions. . Demand: yu 0 . Demand LWP] = g(t): Apply the operator to the guess in the first step, use the demand in the second step, and that Lvi]-0 and L]- 0, and you will get a second condition . Together, in matrix form, the two conditions are . Recall how to inert a 2 by 2 matrix (), and that the determinant of the matrix above is what we call the Wronskian Wy,wl. The result is ngī and th= . Go through the steps in Exanille 1 of Braun on pg、 155 to make sure you can apply this to an easy case.

Answer 1

LCo Lej 9. 또 0 1-92g)