Solve second order DE: $sin^2 xfrac{d^2y}{dx^2} = 2y$

Question

Solve:
$$sin^2 x frac{d^2y}{dx^2} = 2y $$

So what I did was separation of variables, it got me $$frac{y''} {2y}= csc^2 x$$
and integrating both sides will give
$$frac{y'}{2y}= -cot x+ C $$
On another integration we will get
$$frac{ln y}{2} = -ln(sin x) + Cx + K$$
then we get
$$ln y = -2ln(sin x) + 2Cx + 2K$$
and so my answer is
$$y = e^{-2ln(sin x) + 2Cx}$$
but I believe it is wrong.

Z Ahmed · Answer

The ODE is $$frac{d^2y(x)}{dx^2}-2csc^2x~~ y(x)=0$$
$y_1(x)=cot x$ can be checked to satisfy this ODE.
The other solution of this ODE can be found as $y+2(x)=y_1(x) z(x)$,
where $$z(x)=int frac{dx}{y_1^2(x)}-int tan^2 x dx=sec x-x$$
So $$y_2(x)=cot x(tan x-x)=1-xcot x$$
So two linearly independent solutions of the given ODE are:
$$y_1(x)=cot x,~~ y_2(x)=1-x cot x$$

Solve second order DE: $sin^2 xfrac{d^2y}{dx^2} = 2y$

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