  \textbf{4.} Najd�te �e�en� $z = z(x, y)$ rovnice
      $$
            \frac{\partial z}{\partial y} = x^2 + 2y
      $$ spl�uj�c� podm�nku $z(x, x^2) = 1$.

      \textbf{�e�en�}
      $$
            \frac{\partial z}{\partial y} = x^2 + 2y \Rightarrow
            z(x, y) = x^2 y + y^2 + P(x),
      $$ kde $P(x)$ je polynom v $x$. Tvar polynomu ur��me z podm�nky, �e $z(x, x^2) = 1$:
      $$
            z(x, x^2) = x^4 + x^4 + P(x) = 2x^4 + P(x) = 1 \Rightarrow P(x) = -2x^4 + 1,
      $$ tedy $z(x, y) = -2x^4 + x^2 y + y^2 + 1$.
