91Ó°ÊÓ

When tackling a differential equation, the first step is often finding the complementary solution. This is the solution to the associated homogeneous equation, where the non-homogeneous term is set to zero.
For example, given the differential equation \( y'' + 9y = e^x \cos x \), the complementary equation would be \( y'' + 9y = 0 \).
To solve, we suggest a solution of the form \( y = e^{rx} \). Substituting into the homogeneous equation, we find the characteristic equation: \( r^2 + 9 = 0 \). The roots are \( r = \pm 3i \), which indicate a complex scenario, leading to a complementary solution:

• \( y_c = C_1 \cos(3x) + C_2 \sin(3x) \)

Here, \( C_1 \) and \( C_2 \) are constants determined by initial conditions. This part of the solution addresses the oscillatory nature described by the complex roots.

Once the complementary solution is found, the focus shifts to finding a particular solution for the non-homogeneous differential equation. The goal is to find a specific solution that satisfies the entire differential equation.
Based on the method of undetermined coefficients, the form of the particular solution depends on the non-homogeneous term \( e^x \cos x \) itself.

• For \( e^x \cos x \), we propose a particular solution of the form: \( y_p = e^x(A \cos x + B \sin x) \)

The coefficients \( A \) and \( B \) are the unknowns we need to determine.
Through substitution into the original differential equation, derivatives of this trial solution are computed. The product rule plays a critical role here in evaluating these derivatives:

• \( y_p' = e^x(-A \sin x + B \cos x) + e^x(A \cos x + B \sin x) \)
• \( y_p'' = e^x(-A \cos x - B \sin x) + 2e^x(-A \sin x + B \cos x) + e^x(-A \cos x - B \sin x) \)

These expressions are then substituted back into the differential equation to isolate \( A \) and \( B \) by equating coefficients.

A non-homogeneous differential equation is one that includes an external source term, often making the equation more complex to solve.
For an equation like \( y'' + 9y = e^x \cos x \), the right-hand side, \( e^x \cos x \), serves as the non-homogeneous term that affects the form of our solution. In contrast to homogeneous equations where the right-hand side is zero, these non-zero terms require a particular solution.
The general solution to a non-homogeneous differential equation is a combination of its complementary and particular solutions:

• Complementary Solution (\(y_c\)): Resolves the homogeneous part and captures the natural response of the system, not influenced by external forces.
• Particular Solution (\(y_p\)): Account for the non-homogeneous part, the specific influence of external forces on the system.

Adding them gives the full solution:

• \( y = y_c + y_p \)

In our specific problem, this manifests as \( y = C_1 \cos(3x) + C_2 \sin(3x) + e^x \left( \frac{7}{53} \cos x - \frac{2}{53} \sin x \right) \). This ensures the solution satisfies both the intrinsic characteristics of the system and the external excitations described by the right-hand term.