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To solve the differential equation \( y' + (2x - 1)y = 4y \), we first rewrite it in a standard form.

Given:
\[ y' + (2x - 1)y = 4y \]

Subtract \( 4y \) from both sides:
\[ y' + (2x - 1)y - 4y = 0 \]
\[ y' + (2x - 5)y = 0 \]

This is a first-order linear differential equation. The standard form of a first-order linear differential equation is:
\[ y' + p(x)y = 0 \]

Here, \( p(x) = 2x - 5 \).

We solve this using the integrating factor method. The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int (2x - 5) \, dx} \]

Calculate the integral:
\[ \int (2x - 5) \, dx = x^2 - 5x \]

So, the integrating factor is:
\[ \mu(x) = e^{x^2 - 5x} \]

Multiply both sides of the differential equation by the integrating factor:
\[ e^{x^2 - 5x} y' + e^{x^2 - 5x} (2x - 5)y = 0 \]

Notice that the left-hand side is the derivative of \( y \cdot e^{x^2 - 5x} \):
\[ \frac{d}{dx} \left( y \cdot e^{x^2 - 5x} \right) = 0 \]

Integrate both sides with respect to \( x \):
\[ y \cdot e^{x^2 - 5x} = C \]

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

Solve for \( y \):
\[ y = C \cdot e^{-(x^2 - 5x)} \]
\[ y = C \cdot e^{5x - x^2} \]

Therefore, the solution to the differential equation is:
\[ y = C \cdot e^{5x - x^2} \]

Would you like more details on any part of this solution or have any questions? Here are 5 questions you might ask next:
1. How do we determine the integrating factor for a first-order linear differential equation?
2. What are the steps involved in solving first-order linear differential equations?
3. Can you provide examples of other types of differential equations and their solutions?
4. How can we verify the solution to a differential equation?
5. What are some applications of differential equations in real-world problems?

**Tip:** When solving first-order linear differential equations, using the integrating factor method is a powerful technique. Make sure to carefully calculate the integrating factor and apply it correctly to simplify the equation for integration.

Learn how to solve first-order linear differential equations using the integrating factor method with a detailed step-by-step solution. This problem covers key concepts in differential equations, including the integrating factor formula, derivative rules, and integral rules. Suitable for advanced high school or college-level students.

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