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How to solve this differentiation equation? $\sin^2 x {d^2y \over dx^2} = 2 y$ I don't know how to begin. Can it be any simpler than this?

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Cleaning up Maple's solution, I get

$ y \left( x \right) ={\frac {c_{{1}} \left( \cos \left( 2\,x \right) +1 \right) }{\sin \left( 2\,x \right) }}+{\frac {c_{{2}} \left( x\cos \left( 2\,x \right) -\sin \left( 2\,x \right) +x \right) }{\sin \left( 2\,x \right) }} $

I can't imagine that anyone would assign that differential equation as homework to be solved by hand. Are you sure the assignment requires solving the differential equation in closed form?

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    Sorry, I meant reduction of order. If $y(x) = u(x) \cot x$, the differential equation becomes $\sin(x) \cos(x) u'' - 2 u' = 0$, which is a first order equation in $u'$.2012-07-29