1. The problem involves solving and analyzing the given differential expressions and their integrals. 2. For the first equation, $y'_1 = - e^{2x}$, integrating both sides with respect to $x$ gives: $$\int dy = \int - e^{2x} dx \implies y = - \frac{e^{2x}}{2} + C_1$$ 3. For the second equation, $y'_2 = 1$, integrating both sides with respect to $x$ gives: $$\int dy = \int dx \implies y = x + C_2$$ 4. The constants of integration are expressed as: $$C_1 = y + \frac{e^{2x}}{2}$$ $$C_2 = y - x$$ 5. The problem also involves derivatives and mixed partial derivatives such as: $$x = e^{2x}, \quad y = 1, \quad x_x = 2 e^{2x}, \quad y_y = 0, \quad x_y = 0$$ 6. Higher order derivatives and mixed derivatives are given: $$x_{xx} = 2 e^{2x}, \quad y_{yy} = 0, \quad x_y = y_x = 0$$ 7. The expressions involving $U_x$, $U_y$, $U_{xx}$, and $U_{gg}$ are: $$- U_x = e^{2x} V_{13} - V_{12}$$ $$U_y = V_{53} + V_{12}$$ $$U_{xx} = e^{4x} V_{13} - 2 e^{2x} V_{12} + V_{12} + 2 e^{2x} V_{13}$$ $$U_{gg} = V_{33} + 2 V_{13} + V_{22}$$ 8. Combining terms leads to the equation: $$e^{4x} V_{13} - 2 e^{2x} V_{12} + V_{12} + 2 e^{4x} V_{13} + e^{2x} V_{53} + 2 e^{2x} V_{13} + e^{2x} V_{11} + y V_{13} + y V_{12} - x e^{2x} V_{13} - V_{12} = 0$$ 9. Grouping terms by $V_{ij}$ and simplifying: $$V_{13} (e^{4x} + e^{2x}) + V_{22} (1 + e^{2x}) - V_1 (2 e^{2x} - x e^{2x} y) + V_1 (y + x) = 0$$ This summarizes the given expressions and their relationships. Final answer: The integral solutions for $y$ are $$y = - \frac{e^{2x}}{2} + C_1$$ and $$y = x + C_2$$ with the given derivative relations and combined expressions as above.