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We are asked to solve the initial value problem (IVP) given by the differential equation:

\[
\frac{dy}{dx} = 4xy + 1x - 24y - 6, \quad y(0) = \frac{2}{4}.
\]

This is a first-order linear non-homogeneous differential equation. Let's break down the steps to solve it:

### Step 1: Rewrite the Equation

Rearrange the terms of the differential equation:

\[
\frac{dy}{dx} + (24y - 4xy) = x - 6.
\]

It seems this is not immediately separable or an exact equation, so the next step will involve simplifying it by checking for possible integrating factors or solution methods such as substitution, if necessary. 

### Step 2: Initial Condition

We also have the initial condition \( y(0) = \frac{2}{4} = 0.5 \).

I'll solve this equation and get back to you with the explicit solution for \( y \).The general solution to the differential equation is:

\[
y(x) = \frac{C_1 e^{2x(x - 12)}}{4} - \frac{1}{4}.
\]

### Step 3: Apply the Initial Condition

We are given the initial condition \( y(0) = 0.5 \). Let's apply this condition to solve for the constant \( C_1 \).

Substitute \( y(0) = 0.5 \) into the solution to find \( C_1 \).Using the initial condition \( y(0) = 0.5 \), we find that \( C_1 = 3 \).

### Final Solution:
The particular solution to the initial value problem is:

\[
y(x) = \frac{3 e^{2x(x - 12)}}{4} - \frac{1}{4}.
\]

Would you like a breakdown of any specific part of the solution? 

Here are 5 related questions to deepen your understanding:
1. How does the initial condition influence the solution?
2. What are the steps for solving a first-order linear differential equation?
3. What are integrating factors, and when are they used in solving differential equations?
4. Can this type of differential equation be solved using numerical methods instead of symbolic ones?
5. How does the exponential term affect the behavior of the solution for large \(x\)?

### Tip:
Always check your solution by substituting it back into the original differential equation to verify its correctness.

Solve the initial value problem: dy/dx = 4xy + 1x - 24y - 6, y(0) = 2/4.

Learn how to solve the first-order linear differential equation dy/dx = 4xy + x - 24y - 6, with the initial condition y(0) = 2/4. This solution involves methods of solving non-homogeneous differential equations and applying initial conditions to find particular solutions. Suitable for undergraduate calculus students.

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