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We are given the following initial value problem to solve:

\[
\frac{dy}{dx} + xy^2 e^{-x} = -xe^{-x}, \quad y > 0, \quad y(0) = 3.
\]

### Step-by-Step Solution

We can solve this using an integrating factor or by attempting to simplify the equation directly.

1. **First, simplify the differential equation:**

   Move all terms involving \(y\) and \(\frac{dy}{dx}\) to one side:
   \[
   \frac{dy}{dx} = -xe^{-x} - xy^2 e^{-x}.
   \]
   Factor out the common term \(e^{-x}\):
   \[
   \frac{dy}{dx} = -xe^{-x}(1 + y^2).
   \]

2. **Separate the variables:**

   To solve, separate the \(y\)-terms and \(x\)-terms:
   \[
   \frac{dy}{1 + y^2} = -xe^{-x} dx.
   \]

3. **Integrate both sides:**

   The left-hand side can be integrated as an inverse tangent function:
   \[
   \int \frac{dy}{1 + y^2} = \tan^{-1}(y).
   \]

   The right-hand side requires integration by parts. Let’s solve:
   \[
   \int -xe^{-x} dx.
   \]

   Using integration by parts, we set \(u = x\) and \(dv = e^{-x} dx\). Then \(du = dx\) and \(v = -e^{-x}\). Applying the integration by parts formula:
   \[
   \int -xe^{-x} dx = -xe^{-x} + \int e^{-x} dx = -xe^{-x} - e^{-x}.
   \]

   So, the integral of the right-hand side is:
   \[
   \int -xe^{-x} dx = -e^{-x}(x + 1).
   \]

4. **Write the integrated equation:**

   Now we have the integrated equation:
   \[
   \tan^{-1}(y) = -e^{-x}(x + 1) + C,
   \]
   where \(C\) is the constant of integration.

5. **Apply the initial condition:**

   Use the initial condition \(y(0) = 3\) to solve for \(C\). Substituting \(x = 0\) and \(y = 3\) into the equation:
   \[
   \tan^{-1}(3) = -e^{0}(0 + 1) + C,
   \]
   \[
   \tan^{-1}(3) = -1 + C,
   \]
   \[
   C = \tan^{-1}(3) + 1.
   \]

6. **Final solution:**

   The final solution is:
   \[
   \tan^{-1}(y) = -e^{-x}(x + 1) + \tan^{-1}(3) + 1.
   \]

To express \(y\) explicitly, take the tangent of both sides:
\[
y = \tan\left(-e^{-x}(x + 1) + \tan^{-1}(3) + 1\right).
\]

---

Would you like further details on any of the steps or clarifications?

### Related Questions:
1. What is the general method for solving first-order separable differential equations?
2. Can you explain the integration by parts technique in more detail?
3. How do you find integrating factors for linear first-order differential equations?
4. What are the conditions for the existence and uniqueness of solutions for initial value problems?
5. How can inverse trigonometric functions help in solving differential equations?

### Tip:
Always check if a differential equation can be simplified or transformed using known techniques like separation of variables or integrating factors.

Solve the initial value problem: \( \frac{dy}{dx} + xy^2 e^{-x} = -xe^{-x}, y > 0, y(0) = 3 \).

Learn how to solve an initial value problem involving a first-order differential equation with separable variables. This step-by-step solution includes the use of integration by parts, inverse tangent integration, and applying the initial condition y(0) = 3.

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