d/dx (y) = e ^ (3x - 2y) + x ^ 2 * e ^ (-2y)

To solve the differential equation \(\frac{d}{dx}(y) = e^{3x - 2y} + x^2 e^{-2y}\), we need to recognize that this is a first-order differential equation in \(y(x)\).

### Step 1: Rewrite the equation
\[
\frac{dy}{dx} = e^{3x - 2y} + x^2 e^{-2y}
\]

### Step 2: Attempt to separate variables
This differential equation cannot be easily separated into functions of \(x\) and \(y\) alone. Therefore, standard methods like separation of variables do not apply directly. However, we can explore other methods like substitution or an integrating factor.

### Step 3: Substitution to simplify the equation
Let's try a substitution that could simplify the equation. A good candidate might be to let \(u = -2y + \ln(e^{3x} + x^2)\). However, this is still complex. Given this complexity, we could also try solving the equation numerically or applying a method specific to nonlinear differential equations, such as finding an exact solution or using numerical methods.

Would you like to explore specific methods further, or try a numerical approach?

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**Further Questions:**

1. Would you like to see how to apply an integrating factor to solve a similar differential equation?
2. Are you interested in exploring numerical methods to approximate the solution?
3. Would you like a detailed explanation of the substitution method for nonlinear differential equations?
4. Do you want to know how to check the solution of a differential equation?
5. Would you like to discuss exact solutions for specific types of first-order differential equations?

**Tip:** When solving differential equations, identifying the type of equation (e.g., linear, separable, exact) is crucial for selecting the most efficient method of solution.

Explore how to solve the first-order differential equation d/dx(y) = e^(3x - 2y) + x^2 * e^(-2y) step-by-step. Learn methods like substitution and integrating factor to find solutions. Suitable for advanced college-level students and professionals in mathematics and engineering.

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