ScalingStacks

Proof. [022X]

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Proof.

Observe

(v′v2/(n+2))′=v​v′′−2n+2​v′2v(n+4)/(n+2).\left(\frac{v^{\prime}}{v^{2/(n+2)}}\right)^{\prime}=\frac{vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2}}{v^{(n+4)/(n+2)}}.

We can rewrite the ODE (7) as

v​v′′−2n+2​v′2v(n+4)/(n+2)(n+2nvn/(n+2)+(1−t)v′v2/(n+2))n−2=const⋅v−3n/(n+2).\frac{vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2}}{v^{(n+4)/(n+2)}}\left(\frac{n+2}{n}v^{n/(n+2)}+(1-t)\frac{v^{\prime}}{v^{2/(n+2)}}\right)^{n-2}=\text{const}\cdot v^{-3n/(n+2)}.

Now

w=n+2n​vn/(n+2),w′=v′v2/(n+2),w′′=v​v′′−2n+2​v′2v(n+4)/(n+2),w=\frac{n+2}{n}v^{n/(n+2)},\quad w^{\prime}=\frac{v^{\prime}}{v^{2/(n+2)}},\quad w^{\prime\prime}=\frac{vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2}}{v^{(n+4)/(n+2)}},

so the ODE simplifies to (10) after slightly modifying the constant. ∎

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