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


Things will look nicer if we let $y=w^2$. Then $\frac{dy}{dx}=2w\frac{dw}{dx}$ and we end up with $$2w\frac{dw}{dx}+\frac{w^2}{x-2}=5(x-2)w.$$ There is the solution $w=0$. For others, cancel. We get a nice linear equation. The $x-2$ is slightly annoying, at least for typing, so let $t=x-2$. We have arrived at $$2\frac{dw}{dt}+\frac{w}{t}=5t.\tag{$1$}$$

The homogeneous equation $\frac{2dw}{dt}+\frac{w}{t}=0$ is easy to solve. For a particular solution of $(1)$, look for a solution of shape $at^2$.

  • $\begingroup$ also set $v = x-2$ so it is $2\frac{dw}{dv}+\frac{w}{v}=5*v$ $\endgroup$ – kaine Apr 19 '13 at 17:24
  • $\begingroup$ @kaine: We had a similar wish to make things look cleaner. $\endgroup$ – André Nicolas Apr 19 '13 at 17:26
  • $\begingroup$ Alternatively to finding a particular solution to $(1)$, one could multiply through by the integrating factor $\sqrt{t}/2$. $\endgroup$ – preferred_anon Apr 19 '13 at 18:25

$$\frac{dy}{dx}+\frac{y}{x-2}=5(x-2)\sqrt{y}$$ 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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