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- Thread starter suvadip
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- #2

- Jan 26, 2012

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P^2=(dy/dx)^2

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- #4

- Jan 26, 2012

- 4,191

So, in that case, one thing you can try is simply solve for the derivative algebraically first:P^2=(dy/dx)^2

\begin{align*}

3xy&=2x^{2} \frac{dy}{dx}-2 \left( \frac{dy}{dx} \right)^{ \! 2} \\

0&=2 \left( \frac{dy}{dx} \right)^{ \! 2}-2x^{2} \frac{dy}{dx}+3xy \\

\frac{dy}{dx} &= \frac{2x^{2} \pm \sqrt{4x^{4}-4(8)(3xy)}}{4} \\

&= \frac{x^{2} \pm \sqrt{x^{4}-24xy}}{2}.

\end{align*}

Unfortunately, either of the resulting DE's,

$$ \frac{dy}{dx}=\frac{x^{2} + \sqrt{x^{4}-24xy}}{2}$$

or

$$ \frac{dy}{dx}=\frac{x^{2} - \sqrt{x^{4}-24xy}}{2}$$

seem rather forbidding. They might be homogeneous, though. Try that.

[EDIT] Never mind about the homogeneous bit. Neither resulting DE is homogeneous.

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- #5

- Jan 26, 2012

- 4,191

{\frac{-\left( e^{\frac{3\,C(1)}{2}}\,

\left( 3\,e^{\frac{3\,C(1)}{2}} -

{\sqrt{6}}\,x^{\frac{3}{2}} \right) \right) }{3}}}\} ,

\{ {{y(x)}\rightarrow

{\frac{-\left( e^{\frac{3\,C(1)}{2}}\,

\left( 3\,e^{\frac{3\,C(1)}{2}} +

{\sqrt{6}}\,x^{\frac{3}{2}} \right) \right) }{3}}}\} \}.$$

Also, by inspection, you can see that $y=0$ solves the DE. From the looks of the solutions above, this might be a singular solution.

- Jan 26, 2012

- 183

Divide your equation by $x$ and then differentiate. You should find that the new equation factors into two pieces that integrate easily. With these, go back to the original equation and check that they both work (and adjust constants accordingly).

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