Proof. [022C]

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Proof.

We compute

\frac{\partial u}{\partial x_{1}}=(\frac{n+2}{n}v-tv^{\prime})x_{1}^{2/n},\quad\frac{\partial u}{\partial x_{2}}=\frac{d_{1}}{d_{2}}v^{\prime}x_{1}^{2/n},

and the second derivatives

\begin{cases}&\frac{\partial^{2}u}{\partial x_{1}^{2}}=x_{1}^{2/n-1}\{\frac{2(n+2)}{n^{2}}v-\frac{4t}{n}v^{\prime}+t^{2}v^{\prime\prime}\},\\ &\frac{\partial^{2}u}{\partial x_{1}\partial x_{2}}=\frac{d_{1}}{d_{2}}(\frac{2}{n}v^{\prime}-tv^{\prime\prime})x_{1}^{\frac{2}{n}-1},\\ &\frac{\partial^{2}u}{\partial x_{2}^{2}}=(\frac{d_{1}}{d_{2}})^{2}x_{1}^{\frac{2}{n}-1}v^{\prime\prime}.\end{cases}

(8)

Whence

\det(D^{2}u)=(\frac{d_{1}}{d_{2}})^{2}x_{1}^{\frac{4}{n}-2}(\frac{2(n+2)}{n^{2}}vv^{\prime\prime}-\frac{4}{n^{2}}v^{\prime 2}),

d_{1}\frac{\partial u}{\partial x_{1}}+d_{2}\frac{\partial u}{\partial x_{2}}=d_{1}x_{1}^{2/n}(\frac{n+2}{n}v+(1-t)v^{\prime}),

so the NA MA equation becomes

(\frac{2(n+2)}{n^{2}}vv^{\prime\prime}-\frac{4}{n^{2}}v^{\prime 2})(\frac{n+2}{n}v+(1-t)v^{\prime})^{n-2}=\text{const}.

∎