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The differential equation given is:

\[
xy' + 4y = 3x^2
\]

with the initial condition \( y(1) = 9 \).

Let's solve this step-by-step:

### Step 1: Rewrite the equation
The equation is linear in standard form. Dividing the entire equation by \(x\), we get:

\[
y' + \frac{4}{x}y = 3x
\]

This is now a first-order linear differential equation.

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is found by solving:

\[
\mu(x) = e^{\int \frac{4}{x} dx} = e^{4 \ln|x|} = |x|^4
\]

Since we are interested in positive \(x\), we can drop the absolute value:

\[
\mu(x) = x^4
\]

### Step 3: Multiply the equation by the integrating factor
Multiplying both sides of the equation \( y' + \frac{4}{x}y = 3x \) by \(x^4\), we get:

\[
x^4 y' + 4x^3 y = 3x^5
\]

### Step 4: Simplify to a product rule
The left-hand side is the derivative of \(x^4 y\):

\[
\frac{d}{dx}(x^4 y) = 3x^5
\]

### Step 5: Integrate both sides
Integrating both sides with respect to \(x\):

\[
x^4 y = \int 3x^5 dx = \frac{3x^6}{6} = \frac{x^6}{2}
\]

### Step 6: Solve for \(y\)
Now, divide by \(x^4\):

\[
y = \frac{x^6}{2x^4} + \frac{C}{x^4} = \frac{x^2}{2} + \frac{C}{x^4}
\]

### Step 7: Apply the initial condition
We are given that \( y(1) = 9 \). Substituting \(x = 1\) and \(y = 9\) into the equation:

\[
9 = \frac{1^2}{2} + \frac{C}{1^4} = \frac{1}{2} + C
\]

Solving for \(C\):

\[
9 - \frac{1}{2} = C \implies C = \frac{17}{2}
\]

### Step 8: Final solution
Thus, the solution to the differential equation is:

\[
y(x) = \frac{x^2}{2} + \frac{17}{2x^4}
\]

Let me know if you have any questions or would like further details.

Here are 5 related questions:
1. What are first-order linear differential equations?
2. How do you find the integrating factor for a linear differential equation?
3. Why do we divide the original equation by \(x\) to simplify it?
4. Can this method be applied to non-linear differential equations?
5. How does the initial condition help in finding the particular solution?

**Tip:** When solving first-order linear differential equations, always look for the integrating factor to simplify the process.

Solve the differential equation xy' + 4y = 3x^2 with initial condition y(1) = 9.

Learn how to solve the first-order linear differential equation xy' + 4y = 3x^2 with initial condition y(1) = 9. This solution demonstrates the use of the integrating factor method and step-by-step procedures to find the general and particular solution.

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