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Hello Sun,

You have actually gotten most of the way there, keep pushing forward.  This question is of exactly the same type as a previous one I answered here yesterday, it is an example of an Euler equation. 

With the roots r in hand, you can write down the general solution immediately.  In this case, since the roots r are complex conjugates, let me denote them by a ± bi, the solution is of the form

y = |x|a ( c1 cos(b ln |x|) + c2 sin(b ln |x|) ).

You were given an initial value problem to solve, so you would determine the constants ci by applying the initial conditions, giving

c1 = 2 and c2 = -1.

Incidentally, it would not be a bad idea to graph the solution and observe how it behaves.  You could also learn a lot by changing the initial values and plotting a representative selection of solution curves.  Especially note how the solutions behave as x approaches 0, which is the singular point for this equation.


Hassan H.