91Ó°ÊÓ

In the context of solving differential equations, a homogeneous solution refers to solving the associated homogeneous equation. For the example provided, the homogeneous differential equation is \( y'' + 4y = 0 \). This equation does not include any external forcing terms, meaning the right-hand side is equal to zero. The characteristic equation, derived from this homogeneous differential equation, is \( r^2 + 4 = 0 \). From this, we find the roots \( r = \pm 2i \). This suggests the presence of imaginary roots, indicative of oscillatory solutions. As a result, the homogeneous solution takes the form:

• \( y_h(t) = c_1 \cos(2t) + c_2 \sin(2t) \)

The coefficients \( c_1 \) and \( c_2 \) are constants that will be determined by initial or boundary conditions.

Finding a particular solution involves solving the non-homogeneous part of the differential equation. This is achieved using a technique specific to the equation and the form of the non-homogeneous term. In this example, we are tasked with solving \( y'' + 4y = -2 \). The right-hand side \(-2\) is a constant, which simplifies the process. A suitable guess here is a constant function:

• Assume \( y_p(t) = A \)

Upon substituting into the differential equation \( y_p'' + 4y_p = -2 \), we equate the result to \(-2\), solving for \( A \). This leads to \( 4A = -2 \) or \( A = -\frac{1}{2} \). Therefore, the particular solution is:

• \( y_p(t) = -\frac{1}{2} \)

This function encompasses the forced response of the system to the external input.

The method of undetermined coefficients is a powerful technique for finding particular solutions to linear non-homogeneous differential equations. It works particularly well with equations where the non-homogeneous part (right-hand side) is a polynomial, exponential, sine, cosine, or their combinations.
In this method, we assume a form for the particular solution that mirrors the form of the non-homogeneous term. In our example, since the term \(-2\) is constant, we assumed \( y_p(t) = A \). The goal is to substitute this assumed form back into the differential equation and solve for the unknown coefficients (in our example, just \( A \)).

• Substitute \( y_p(t) = A \) into \( y'' + 4y = -2 \)
• Equate and solve for \( A \)
• Obtain \( A = -\frac{1}{2} \)

Thus, the undetermined coefficients have been determined, providing the particular solution.

A system of equations arises when applying initial or boundary conditions to determine the constants in the general solution of a differential equation. In our problem, once the homogeneous and particular solutions were combined, forming \( y(t) = c_1 \cos(2t) + c_2 \sin(2t) - \frac{1}{2} \), initial conditions were applied:

• \( y(\pi / 8) = \frac{1}{2} \)
• \( y'(\pi / 8) = 2 \)

By substituting these conditions into the general solution and its derivative, we derive a system of linear equations involving \( c_1 \) and \( c_2 \). Solving this system involves:

• Equation 1: \( c_1 + c_2 = \sqrt{2} \)
• Equation 2: \( -c_1 + c_2 = 2 \)

By adding and subtracting these equations, we can independently solve for each constant. This process transforms abstract constants into precise values, bringing us closer to the specific solution that fits the initial conditions.