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

[tex] yy''=y'^{2}-y'^{3} [/tex]

I'm quite sure I got lost somewhere. Can anyone show me where?

Thanks

Set

[tex]z(y)=y' [/tex]

then

[tex]\frac{\partial z}{\partial y}=y''\cdot y'=zy'' so y''=\frac{z'}{z} [/tex]

Plugging this in

[tex] y\frac{z'}{z}=z^{2}\left(1-z\right) and so \frac{1}{y}\partial y=\frac{\partial z}{z^{3}\left(1-z\right)} [/tex]

Integrating we have

[tex] ln\left(y\right)=\int\frac{1}{z^{3}}+\frac{1}{z^{2}}+\frac{1}{z}-\frac{1}{z-1}=-\frac{1}{z^{2}}-\frac{1}{z}+ln\left(\frac{z}{z-1}\right)+c [/tex]

So

[tex] y=c_{1}\left(\frac{z}{z-1}\right)e^{-\left(\frac{z+1}{^{z^{2}}}\right)} [/tex] and recalling that z=y' we have

[tex] y=c_{1}\left(\frac{y'}{y'-1}\right)e^{-\left(\frac{y'+1}{y'^{2}}\right)} [/tex]

What now?

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Seconed order non linear ODE

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