Given the equation:

$\frac{dy}{dx} = e^{2x - 3y} + 4x^2 e^{-3y}$

This is a first-order differential equation, and we can attempt to solve it or simplify it.

Step-by-Step Solution Approach:

1. Rewrite the equation: $\frac{dy}{dx} = e^{2x - 3y} + 4x^2 e^{-3y}$ It is not immediately separable, so finding a particular solution or trying to simplify further might be a good approach.

2. Separate terms: While this equation involves both $x$ and $y$, it is difficult to separate them because of the exponential terms involving both $x$ and $y$.

3. Substitution: You may attempt a substitution to reduce the complexity of the equation, but first, it helps to analyze the equation’s form to see if it resembles any standard methods, such as:

   • Exact equations
   • Integrating factors
   • Homogeneous equations

Would you like to pursue a specific method, such as attempting a substitution for $u = e^{-3y}$, or see a detailed solution?

---

Related Questions:

1. What methods can be used to solve first-order differential equations?
2. How does substitution help simplify complex differential equations?
3. What is an integrating factor in solving differential equations?
4. How are separable differential equations different from non-separable ones?
5. Can exponential functions involving multiple variables be simplified for easier integration?

Tip: Always check if a given differential equation can be separated before applying more advanced methods like substitutions or integrating factors.