The indicated function y1(x is a solution of the given differential equation. use reduction of order or formula (5 in section 4.2, y2 = y1(x e? ? p(x dx y 2 1 (x dx (5 as instructed, to find a second solution y2(x. xy'' y' = 0; y1 = ln x

Given a solution , we can attempt to find a solution of the form . We have derivatives





Substituting into the ODE, we get




Setting , we end up with the linear ODE



Multiplying both sides by , we have



and noting that



we can write the ODE as



Integrating both sides with respect to , we get




Now solve for :




So you have



and given that , the second term in is already taken into account in the solution set, which means that , i.e. any constant solution is in the solution set.

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