91Ó°ÊÓ

In the context of differential equations, a particular integral is a specific solution to a non-homogeneous differential equation. This integral differs from the complementary solution, which relates only to the homogeneous components of the equation. The process of finding a particular integral involves seeking a function that satisfies the entire non-homogeneous equation.

For the given problem, the equation is \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}}-3 \frac{\mathrm{d} x}{\mathrm{~d}t}+2 x=5 \mathrm{e}^{3 t}\). To find a particular integral, we aim to locate a function that directly addresses the non-homogeneous part, here represented by \(5e^{3t}\). While the complementary solution tackles the equation without a non-homogeneous term, the particular integral requires focusing on the differentials and constants present in the complete form of the equation.

The systematic approach involves:

• Identifying the type of non-homogeneous term.
• Proposing a trial solution that reflects the structure of this term.
• Substituting the trial solution into the differential equation.
• Solving for any unknown constants within the trial solution.

By following this methodology, you determine the particular integral that complements the homogeneous solution, providing a complete solution to the differential equation.

A non-homogeneous differential equation includes an extra term not present in its homogeneous counterpart. This term is what principally drives the need for a particular integral. In our exercise, the equation is given as \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}} - 3 \frac{\mathrm{d} x}{\mathrm{~d}t} + 2x = 5 \mathrm{e}^{3t}\).

The non-homogeneous part, \(5e^{3t}\), sets it apart from the homogeneous equation \(\frac{\mathrm{d}^{2} x}{\mathrm{~d} t^{2}} - 3 \frac{\mathrm{d} x}{\mathrm{~d}t} + 2x = 0\). This type of equation consists of two main sections:

• The homogeneous solution: This utilizes the characteristic equation derived from setting the differential equation equal to zero.
• The particular solution: Focuses on solving the specific non-homogeneous function, \(5e^{3t}\) in this case.

Understanding these components is crucial because each fulfills a distinct role in fully solving the equation. The non-homogeneous term often resembles functions like polynomials, exponential, or trigonometric terms, dictating the form of the trial solution for the particular integral.

The choice of a trial solution is critical in solving for the particular integral in non-homogeneous differential equations. In this specific exercise, the non-homogeneous part \(5e^{3t}\) suggests using an exponential function as a part of the trial solution.

The trial solution proposed is \(x_p(t) = Ae^{3t}\), where \(A\) is a constant that we need to determine. By comparing the trial solution to the structure of the non-homogeneous term, we ensure the solution aligns with the nature of the differential equation. Here's how the process works:

• Use \(x_p(t) = Ae^{3t}\) as a trial solution.
• Calculate the first derivative \(\frac{\mathrm{d} x_p}{\mathrm{d}t} = 3Ae^{3t}\) and second derivative \(\frac{\mathrm{d}^2 x_p}{\mathrm{d}t^2} = 9Ae^{3t}\).
• Substitute these derivatives into the differential equation.
• Identify and simplify terms to solve for the constant \(A\).

After substitution, simplification yielded \(2Ae^{3t} = 5e^{3t}\), leading to \(A = \frac{5}{2}\). Thus, the particular solution becomes \(x_p(t) = \frac{5}{2}e^{3t}\). This approach ensures that our solution fits both the structure and the specific behavior dictated by the non-homogeneous term. By matching the form and solving for \(A\), we address the complexity of the equation effectively.