<s>[INST] Manipulating an ODE into a full differential Find the general solution of:
$$(x-2)\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + \frac{4y}{x^2} = 0$$
If I first multiply through by $x^2$ then I can write the above as
$$x^2(x-2)\frac{d^2y}{dx^2} + 3x^2\frac{dy}{dx} + 4y  $$
$$ =\left[ x^3\frac{d^2y}{dx^2} + 3x^2\frac{dy}{dx}\right]-2x^2 \frac{d^2y}{dx^2} +4y  $$
$$ =\frac{d}{dx}\left( x^3\frac{dy}{dx}\right) -2x^2 \frac{d^2y}{dx^2} +4y = 0$$
and I'm not sure how to express the rest of the equation as the derivative of some function (I tried $\frac{d}{dx} 2x^2y$ to no avail) . Any help would be appreciated.
 [/INST] Hint:
$$\frac{d}{dx}\left( x^3\frac{dy}{dx}\right) -2x^2 \frac{d^2y}{dx^2} +4y = 0$$
$$\frac{d}{dx}\left( x^3\frac{dy}{dx}\right) -2x^2 \frac{d^2y}{dx^2} \color {red}{-4xy'+4xy'} +4y = 0$$
$$\left( x^3y'\right)' -(2x^2 y')'  +4(xy)' = 0$$
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