When we were (re)introduced to the following Cauchy problem:

(i) $$ u_{tt} - u_{xx} = 0 $$

(ii) $$ u|_{t =\frac{x^2}{2}} = x^3 $$

(iii) $$ u_{t} |_{t = \frac{x^2}{2}} = 2x $$

And we took the general solution of

(from i: 1) $$ u(x,t) = \phi(x +t) + \psi (x-t) $$

and applying the conditions (ii,iii)

(from ii: 2) $$ u(x, \frac{x^2}{2}) = \phi(x + \frac{x^2}{2}) + \psi(x - \frac{x^2}{2}) = x^3 $$

(from iii : 3) $$ u_{t} (x,\frac{x^2}{2}) = \phi'(x + \frac{x^2}{2}) - \psi'(x - \frac{x^2}{2}) = 2x $$

(2') $$3x^2 = (1+x) \phi'(x + \frac{x^2}{2}) + (1-x)\psi'(x - \frac{x^2}{2})$$

But from there we wrote that :

$$

\phi'(x+ \frac{x^2}{2}) = \frac{\begin{vmatrix}

2x & -1 \\

3x^2 & 1-x

\end{vmatrix}}{\begin{vmatrix}

1 & -1 \\

1+x & 1-x

\end{vmatrix}} = \frac{2x+ x^2}{2} = x+ \frac{x^2}{2}

$$

and similarly for $$\psi'(x - \frac{x^2}{2}) = -x + \frac{x^2}{2} $$

Where did these determinants come from?