To solve the differential equation

\[
4 \frac{dy}{dx} + 28y = 7,
\]

we can follow these steps:

### Step 1: Simplify the equation
First, divide the entire equation by 4 to simplify:

\[
\frac{dy}{dx} + 7y = \frac{7}{4}.
\]

This is now a first-order linear differential equation of the form:

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

where \(P(x) = 7\) and \(Q(x) = \frac{7}{4}\).

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

\[
\mu(x) = e^{\int P(x)\, dx} = e^{\int 7\, dx}.
\]

Calculate the integral:

\[
\mu(x) = e^{7x}.
\]

### Step 3: Multiply the entire differential equation by the integrating factor
Multiply both sides by \(e^{7x}\):

\[
e^{7x} \frac{dy}{dx} + 7e^{7x}y = \frac{7}{4}e^{7x}.
\]

The left-hand side is the derivative of the product \(y e^{7x}\):

\[
\frac{d}{dx} \left( y e^{7x} \right) = \frac{7}{4} e^{7x}.
\]

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

\[
y e^{7x} = \int \frac{7}{4} e^{7x} \, dx.
\]

To solve the integral on the right-hand side:

\[
\int \frac{7}{4} e^{7x} \, dx = \frac{7}{4} \cdot \frac{1}{7} e^{7x} + C = \frac{1}{4} e^{7x} + C,
\]

where \(C\) is the constant of integration.

### Step 5: Solve for \(y\)
Finally, solve for \(y(x)\) by dividing both sides by \(e^{7x}\):

\[
y(x) = \frac{1}{4} + Ce^{-7x}.
\]

So the general solution to the differential equation is:

\[
\boxed{y(x) = \frac{1}{4} + Ce^{-7x},}
\]

where \(C\) is an arbitrary constant.

Would you like more details on any step? Or do you have any further questions?

---

1. What is the role of the integrating factor in solving differential equations?
2. How do you determine the form of the general solution for a linear differential equation?
3. Can you explain why dividing by the integrating factor simplifies the differential equation?
4. What are the characteristics of a first-order linear differential equation?
5. How would the solution change if the equation had a different constant instead of 7?

**Tip:** When solving linear differential equations, always start by simplifying the equation as much as possible. This often makes finding the integrating factor and the solution easier.

Learn how to solve the differential equation 4(dy/dx) + 28y = 7 with detailed steps. This problem covers first-order linear differential equations and integrating factors, essential topics in advanced calculus and differential equations. Suitable for advanced undergraduate students and enthusiasts of mathematical analysis.

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