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A non-homogeneous differential equation is a type of differential equation that includes a non-zero term on one side known as the non-homogeneous term, or the forcing function. In these equations, the left side typically involves derivatives of a function, while the right side is a function of the independent variable, such as in this example, the equation \[ y'' + 2y' - 3y = 1 + xe^x \]. This equation is non-homogeneous due to the presence of the term \( 1 + xe^x \), which is not equal to zero.

Unlike homogeneous differential equations, where the solution can be expressed as a linear combination of basic solutions due to the right-hand side being zero, non-homogeneous equations require us to find both a complementary solution and a particular solution.

Understanding the differences is essential:

• Homogeneous Equations: All terms are purely in terms of the function and its derivatives.
• Non-homogeneous Equations: Involve additional terms independent from the solution’s function.

These additional terms are the key focus when identifying a particular solution for the equation.

The particular solution of a non-homogeneous differential equation is a specific solution that addresses the non-homogeneous part of the equation. For the given exercise, the solution comprises finding individual particular solutions to each component of the non-homogeneous term \(1 + xe^x\), and then combining them.

You can use these steps to find a particular solution:

• Identify different parts of the non-homogeneous term. Here, they are the constant 1 and the term \(xe^x\).
• Assume a form for the particular solution tailored to each part. For example, assume a constant form \(y_{p1} = A\) for 1, and a form including the similar structures \((Bx + C)e^x\) for \(xe^x\).
• Determine the constants by substituting back into the equation, ensuring the left-hand side equates to the exact non-homogeneous term on the right.

By handling each term separately, you ensure the overall particular solution, \(y_p = -\frac{1}{3} + \frac{1}{2}xe^x\), satisfies the entire equation. This methodical approach allows one to handle each non-constant component effectively.

The method of undetermined coefficients is a popular technique for finding particular solutions of linear non-homogeneous differential equations, especially when the non-homogeneous term has special forms like polynomials, exponentials, or trigonometric functions.

This method involves assuming a form for the particular solution that mimics the non-homogeneous term. Coefficients are initially undetermined and hence the name 'undetermined coefficients'. Here’s how it works:

• Make an educated guess about the form of the particular solution. This guess is often based on the type of the non-homogeneous term.
• In case of duplicates with the complementary solution, multiply the guess with powers of \(x\) until distinct.
• Substitute your guess into the differential equation to find relationships between the coefficients by equating terms.

For example, in the given problem, assuming \(y_{p2} = (Bx + C)e^x\) aligns with the term \(xe^x\). By plugging in these forms and simplifying, coefficients \(B\) and \(C\) are calculated to give the particular solution. Practicing this method equips students with a reliable tool for tackling a range of non-homogeneous differential equations.