https://file.aiquickdraw.com/mathbot/file/1726701571903ietx9fk8.jpg

The equation shown in the image is:

\[
\frac{1}{x} \frac{dy}{dx} - \frac{2}{x^2} y = x e^x
\]

This is a **linear first-order differential equation**. To solve it, we can follow these steps:

### Step 1: Standard form
First, rearrange the equation into the standard form of a linear differential equation:

\[
\frac{dy}{dx} + P(x)y = Q(x)
\]

Multiplying the entire equation by \(x\), we get:

\[
\frac{dy}{dx} - \frac{2}{x} y = x^2 e^x
\]

Now, this is in the standard form with:
- \(P(x) = -\frac{2}{x}\)
- \(Q(x) = x^2 e^x\)

### Step 2: Integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x) dx} = e^{\int -\frac{2}{x} dx} = e^{-2 \ln |x|} = x^{-2}
\]

### Step 3: Multiply through by the integrating factor
Multiply the entire equation by \(x^{-2}\):

\[
x^{-2} \frac{dy}{dx} - \frac{2}{x^3} y = e^x
\]

The left-hand side is the derivative of \(\left( x^{-2} y \right)\):

\[
\frac{d}{dx} \left( x^{-2} y \right) = e^x
\]

### Step 4: Integrate both sides
Integrate both sides with respect to \(x\):

\[
x^{-2} y = \int e^x dx = e^x + C
\]

### Step 5: Solve for \(y\)
Multiply both sides by \(x^2\) to solve for \(y\):

\[
y = x^2 \left( e^x + C \right)
\]

Thus, the general solution to the differential equation is:

\[
y = x^2 e^x + C x^2
\]

This is the solution to the given differential equation.

Would you like further details or have any questions?

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Here are 5 related questions that build on this concept:

1. What is the significance of the integrating factor in solving linear differential equations?
2. How do you determine when a differential equation is linear or non-linear?
3. Can you solve the given equation using a different method, such as variation of parameters?
4. What happens to the solution if the right-hand side is zero (homogeneous equation)?
5. How do you interpret the constant \(C\) in the context of initial conditions?

**Tip:** Always check if your differential equation can be simplified or made into a standard form before attempting to solve it.

Solve the differential equation: (1/x) dy/dx - (2/x^2) y = x e^x.

Explore the solution to the linear first-order differential equation (1/x) dy/dx - (2/x^2) y = x e^x using the integrating factor method. This problem highlights key concepts in calculus and differential equations, suitable for undergraduate students.

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