You have the differential equation:

\[
\frac{dy}{dx} = -2y + 5
\]

This is a first-order linear ordinary differential equation. To solve it, we can use the method of **separation of variables** or **integrating factor**.

Let's proceed with the **integrating factor** method. The equation can be rewritten in the form:

\[
\frac{dy}{dx} + 2y = 5
\]

This is a linear equation of the form:

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

where \(P(x) = 2\) and \(Q(x) = 5\). The solution method involves finding the integrating factor \( \mu(x) \), given by:

\[
\mu(x) = e^{\int P(x) \, dx} = e^{2x}
\]

Multiply the entire differential equation by the integrating factor \(e^{2x}\):

\[
e^{2x} \frac{dy}{dx} + 2 e^{2x} y = 5 e^{2x}
\]

Now, the left side of the equation is the derivative of \( e^{2x} y \):

\[
\frac{d}{dx} \left( e^{2x} y \right) = 5 e^{2x}
\]

Next, integrate both sides with respect to \(x\):

\[
\int \frac{d}{dx} \left( e^{2x} y \right) \, dx = \int 5 e^{2x} \, dx
\]

The integral of the left-hand side is \( e^{2x} y \), and the integral of the right-hand side is:

\[
\int 5 e^{2x} \, dx = \frac{5}{2} e^{2x}
\]

So, we have:

\[
e^{2x} y = \frac{5}{2} e^{2x} + C
\]

Where \(C\) is the constant of integration. Now, divide through by \(e^{2x}\):

\[
y = \frac{5}{2} + Ce^{-2x}
\]

This is the general solution to the differential equation.

Would you like more details on any step, or have any further questions on this? 

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Here are some related questions to explore:
1. How does the integrating factor method work for first-order linear differential equations?
2. What does the constant \(C\) represent in the general solution?
3. How can we solve this type of equation using separation of variables?
4. What happens if the right-hand side of the equation is a more complex function (not just a constant)?
5. Can you solve a similar equation where the coefficient of \(y\) is not constant (e.g., \(\frac{dy}{dx} = -2xy + 5\))?

**Tip**: When solving differential equations, always check if the equation is linear or separable to choose the best solving method.

Learn how to solve the first-order linear differential equation y' = -2y + 5 using the integrating factor method. This step-by-step guide covers key concepts, including the integrating factor theorem and general solution formulation. Suitable for high school and college-level calculus students.

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