Is the following differential equation homogeneous or non-homogeneous?

y'' + 3y' - 2 = 5x + 1

Your written work for this problem is just a quick explanation.

To determine if the differential equation is homogeneous or non-homogeneous, we examine whether there is a non-zero term on the right-hand side of the equation.

The given equation is:
\[
y'' + 3y' - 2 = 5x + 1
\]

For a differential equation to be **homogeneous**, the equation must equal zero, meaning the entire right side should be zero. Here, the right side is \(5x + 1\), which is not zero. Therefore, this equation is **non-homogeneous**.

In summary:
- Since the equation does not equal zero but rather \(5x + 1\), it is a **non-homogeneous differential equation**.

Would you like to see more details on this topic?

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1. What is a homogeneous differential equation in general terms?
2. How would the solution approach differ between a homogeneous and a non-homogeneous differential equation?
3. How do we find the particular solution of a non-homogeneous differential equation?
4. What methods are used to solve non-homogeneous linear differential equations?
5. How do we handle constant coefficients in non-homogeneous differential equations?

**Tip:** Always check the right-hand side of the equation first when determining if it is homogeneous or non-homogeneous.

Is the following differential equation homogeneous or non-homogeneous? y'' + 3y' - 2 = 5x + 1

Learn to identify whether a differential equation is homogeneous or non-homogeneous by examining if the right side of the equation equals zero. In this example, the equation y'' + 3y' - 2 = 5x + 1 has a non-zero right side, indicating it is a non-homogeneous differential equation. Suitable for high school students in Grades 11-12 studying differential equations.

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