The provided image shows a solution to a differential equation that appears to involve separable variables. Let me walk through the solution process as shown in the notes:

The given differential equation is:

$\frac{dy}{du} = e^{3x-2y} + x^2 e^{-2y}$

Steps in the solution:

1. Separate the variables:

   The equation is manipulated to separate terms involving $y$ on one side and terms involving $x$ on the other:

   $\frac{dy}{du} = e^{3x-2y} + x^2 e^{-2y}$ becomes $\frac{dy}{du} = e^{-2y}(e^{3x} + x^2)$ This allows you to handle the terms involving $y$ and $x$ separately.

2. Further separation:

   Multiply both sides of the equation by $e^{2y}$, leading to: $e^{2y} \, dy = (e^{3x} + x^2) \, du$ This separates the variables $x$ and $y$, making it ready for integration.

3. Integrate both sides:

   Integrate both sides of the equation: $\int e^{2y} \, dy = \int (e^{3x} + x^2) \, du$ The integral of the left side is: $\frac{e^{2y}}{2}$ For the right side, apply the formula for exponential and polynomial integration: $\int e^{ax} \, dx = \frac{e^{ax}}{a}, \quad \int x^n \, dx = \frac{x^{n+1}}{n+1}$ So, $\int (e^{3x} + x^2) \, du = \frac{e^{3x}}{3} + \frac{x^3}{3} + C$

4. Combine the results:

   After integrating, the equation becomes: $\frac{e^{2y}}{2} = \frac{e^{3x}}{3} + \frac{x^3}{3} + C$ Multiply both sides by 2 to simplify: $e^{2y} = \frac{2}{3}(e^{3x} + x^3) + 2C$

This gives the general solution to the differential equation.

Additional details or questions:

1. Would you like a detailed explanation of each integration step?
2. Should I explain more about separating variables in differential equations?
3. Do you want me to solve another similar differential equation?
4. Would you like an alternative method for solving this equation?
5. Would you like to discuss how to check if this solution is correct?

Tip:

Always double-check your work by differentiating the final solution to ensure it satisfies the original differential equation.