<s>[INST] Solving a Differential Equation using separation method I have the following problem but I don't know where to start:-
$$\frac{dy}{dx}\ + \frac{y}{x-2}\ = 5(x-2)\sqrt{y} $$
I tried to use the separation method but not able to. Advice/Guidance is much appreciated
 [/INST] $$\frac{dy}{dx}+\frac{y}{x-2}=5(x-2)\sqrt{y}$$
Multiply through by $\frac{1}{2}\sqrt \frac{x-2}{y}$ to get
$$\frac{\sqrt{x-2}}{2\sqrt{y}}\frac{dy}{dx}+\frac{\sqrt{y}}{2\sqrt{x-2}}=\frac{5}{2}(x-2)^{3/2}$$
The LHS is the derivative of $\sqrt{x-2}\sqrt{y}$, so we can integrate:
$$\sqrt{x-2}\sqrt{y}=\int\frac{5}{2}(x-2)^{3/2}dx=\frac{5}{3}(x-2)^{5/2}+A$$
For some arbitrary constant $A$.
Thus, dividing by $\sqrt{x-2}$:
$$\sqrt{y}=\frac{5}{3}(x-2)^{2}+A(x-2)^{-1/2}$$
Then, finally, squaring:
$$y=\frac{25}{9}(x-2)^{4}+\frac{10A}{3}(x-2)^{3/2}+A^{2}(x-2)^{-1}$$
While I think @André's solution is more elegant, this one is perhaps an alternative.
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