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**Web Bonus Problems / Re: Web bonus problem -- Week 2**

« **on:**January 15, 2018, 02:38:01 PM »

Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic systemu=(x+t)

$$\left( \begin{array}{cc|c}

1 & -1 & 2x\\

1+x& 1-x&3x^{2}\\

\end{array} \right)

\implies

\left( \begin{array}{c}

\varphi'(X)\\

\psi'(Y)

\end{array} \right)

=

\left( \begin{array}{c}

X\\

-Y

\end{array} \right)

\text{ where X, Y are the arguments of $\varphi, \psi$ resp. }

\implies

\left( \begin{array}{c}

\varphi(X)\\

\psi(Y)\end{array} \right)

=

\left( \begin{array}{c}

X^{2}/2+C_{1}\\

-Y^{2}/2+C_{2}

\end{array} \right)

\implies

u=(x+t)^{2}/2 - (x-t)^{2}/2 + Cx+D.

$$

The last two terms in the final step are empirical, and I urgently seek a theoretical account for it.

^{2}/2-(x-t)

^{2}/2, where u is define on t>=0 and x is any real number.