(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I must solve [itex]yy''-(y')^2-6xy^2=0[/itex].

2. Relevant equations

Not sure.

3. The attempt at a solution

I reach something but this doesn't satisfy the original DE...

Here is my work:

I divide the DE by [itex]y^2[/itex] to get the new DE [itex]\frac{y''}{y}- \left ( \frac{y'}{y} \right ) ^2-6x=0[/itex]. Now I notice that [itex]\left ( \frac{y'}{y} \right )'=\frac{y''}{y}-1[/itex] so that the DE to solve reduces to [itex]\left ( \frac{y'}{y} \right )'- \left ( \frac{y'}{y} \right ) ^2+1-6x=0[/itex].

This suggests me to call a new variable [itex]v=\frac{y'}{y}[/itex]. Thus the DE to solve reduces to [itex]v'-v^2+1-6x=0[/itex]. It is separable so I'm extremely lucky. I reach that [itex]\ln y = \int (3x^2+c_1)dx+c_2 \Rightarrow y(x)=e^{x^3+c_1x}+c_2[/itex].

Hence [itex]y'=(3x^2+c_1)e^{x^3+c_1x}[/itex] and [itex]6xe^{x^3+c_1x}+(3x^2+c_1)^2e^{x^3+c_1x}[/itex]. Plugging these into the original DE doesn't reduces to 0.

What did I do wrong?

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# Non linear second order DE

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