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.