Toronto Math Forum

Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic system
$$\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.

u=(x+t)²/2-(x-t)²/2, where u is define on t>=0 and x is any real number.

Is it u(x,t)=C₁(x+t)+C₂(x-t). Then I plug in t=x²/2 for u and u_t and I get C₁(x+x²/2)= (x³+x²)/2 and C₂(x-x²/2) = (x³-x²)/2 which is the question i asked before. Can you tell me a bit more? I still don't get it.