The given differential equation is:

\[
Y' = \frac{y}{4} + \frac{3}{2}
\]

This is a first-order linear differential equation. To solve it, we can proceed as follows:

### Step 1: Rewrite the equation

We start by rewriting the equation in standard form for linear differential equations:

\[
Y' - \frac{1}{4}y = \frac{3}{2}
\]

### Step 2: Identify the integrating factor

The integrating factor \( \mu(x) \) is found using the coefficient of \( y \), which is \( -\frac{1}{4} \). The integrating factor is:

\[
\mu(x) = e^{\int -\frac{1}{4}dx} = e^{-\frac{1}{4}x}
\]

### Step 3: Multiply both sides by the integrating factor

Multiplying both sides of the differential equation by \( e^{-\frac{1}{4}x} \):

\[
e^{-\frac{1}{4}x} Y' - \frac{1}{4} e^{-\frac{1}{4}x} y = e^{-\frac{1}{4}x} \cdot \frac{3}{2}
\]

The left-hand side simplifies to the derivative of \( e^{-\frac{1}{4}x} y \):

\[
\frac{d}{dx}\left(e^{-\frac{1}{4}x} y\right) = e^{-\frac{1}{4}x} \cdot \frac{3}{2}
\]

### Step 4: Integrate both sides

Now integrate both sides with respect to \( x \):

\[
e^{-\frac{1}{4}x} y = \int e^{-\frac{1}{4}x} \cdot \frac{3}{2} dx
\]

The integral on the right-hand side is:

\[
\int e^{-\frac{1}{4}x} \cdot \frac{3}{2} dx = -6e^{-\frac{1}{4}x} + C
\]

### Step 5: Solve for \( y \)

Now, solve for \( y \):

\[
e^{-\frac{1}{4}x} y = -6e^{-\frac{1}{4}x} + C
\]

Multiply both sides by \( e^{\frac{1}{4}x} \):

\[
y = -6 + Ce^{\frac{1}{4}x}
\]

### Final Solution:

The general solution to the differential equation is:

\[
y = -6 + Ce^{\frac{1}{4}x}
\]

where \( C \) is an arbitrary constant.

Would you like more details on any of these steps?

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**Relative Questions:**

1. What is the purpose of the integrating factor in solving linear differential equations?
2. How do you compute the integrating factor for first-order linear differential equations?
3. Can you explain why we multiply the entire equation by the integrating factor?
4. How would the solution change if the right-hand side of the equation were different?
5. What happens if the differential equation is homogeneous (i.e., the right-hand side is zero)?

**Tip:** Always check if the differential equation is linear before choosing a method to solve it, as integrating factors are specifically useful for linear equations.

This guide provides a step-by-step solution for solving the first-order linear differential equation Y' = y/4 + 3/2 using integrating factors. Learn the key concepts, including identifying the integrating factor and applying it to solve the equation. Suitable for students in Grades 11-12 or early college-level mathematics.

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