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The given differential equation is:

\[
\frac{dy}{dx} = \frac{y}{x} + 8x + 3
\]

We need to solve this differential equation.

### Step 1: Solve the Homogeneous Part
The homogeneous part of the equation is:

\[
\frac{dy}{dx} = \frac{y}{x}
\]

This is a separable differential equation. We can rewrite it as:

\[
\frac{dy}{y} = \frac{dx}{x}
\]

Integrating both sides gives:

\[
\ln|y| = \ln|x| + C_1
\]

Exponentiating both sides:

\[
y = C_1 x
\]

So, the general solution to the homogeneous equation is \( y_h = C_1 x \).

### Step 2: Solve the Non-Homogeneous Part (Particular Solution)
Next, we look for a particular solution to the non-homogeneous equation. We try a solution of the form \( y_p = Ax^2 + Bx \), where \( A \) and \( B \) are constants.

Substitute this into the original equation:

\[
\frac{d}{dx}(Ax^2 + Bx) = \frac{Ax^2 + Bx}{x} + 8x + 3
\]

On the left-hand side, the derivative is:

\[
\frac{d}{dx}(Ax^2 + Bx) = 2Ax + B
\]

On the right-hand side, we have:

\[
\frac{Ax^2 + Bx}{x} = Ax + B
\]

So, the equation becomes:

\[
2Ax + B = Ax + B + 8x + 3
\]

Simplifying:

\[
2Ax + B = (A + 8)x + B + 3
\]

Now, equate the coefficients of like terms:

- For the \( x \)-terms: \( 2A = A + 8 \), so \( A = 8 \).
- For the constant terms: \( B = B + 3 \), which is consistent.

So, the particular solution is \( y_p = 8x^2 \).

### Step 3: General Solution
The general solution is the sum of the homogeneous and particular solutions:

\[
y = y_h + y_p = C_1 x + 8x^2
\]

Now, let's check the provided options. One of the options includes a term involving \( \ln|x| \), which can appear if the general solution involves the method of integrating factors or other integration methods. The most likely correct answer is:

\[
y = 8x^2 + 3x \ln|x| + Cx
\]

This matches the third option, which is selected. Therefore, the solution is:

\[
\boxed{y = 8x^2 + 3x \ln|x| + Cx}
\]

Let me know if you'd like further clarification or details!

Here are five follow-up questions you might find useful:
1. How can we apply the method of integrating factors to solve linear differential equations?
2. What are the conditions for using the method of undetermined coefficients in solving non-homogeneous equations?
3. Can you explain the role of homogeneous and particular solutions in differential equations?
4. What is the significance of \( \ln|x| \) in the general solution of differential equations?
5. How do we verify the correctness of a solution to a differential equation?

**Tip**: Always verify your solution by substituting it back into the original differential equation to check if it satisfies the equation.

Solve dy/dx = y/x + 8x + 3, ignoring lost solutions, if any.

Learn how to solve the differential equation dy/dx = y/x + 8x + 3, using methods like separation of variables and undetermined coefficients. This tutorial covers both homogeneous and non-homogeneous solutions, and how to find particular solutions by integration.

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