91Ó°ÊÓ

In a differential equation like the one provided, the homogeneous solution is a crucial part of finding the complete solution. For the equation \( \frac{1}{2} y'' + 2y = 0 \), we focus on the part where the equation equals zero. This part is considered the 'homogeneous' part of the differential equation. Here, we ignore the non-homogeneous function \( f(t) \), which in this problem is \( \tan(2t) - \frac{1}{2} e^t \).

For the homogeneous solution \( y_h \), we solve the equation \( \frac{1}{2} y'' + 2y = 0 \) directly.
This involves finding the characteristic equation, which usually gives us insights into the behavior and form of the solutions.

• The focus is on terms involving \( y'' \) and \( y \).
• These terms tell us if the solution will be exponential, trigonometric, or polynomial.

The term 'particular solution' refers to a solution of the original non-homogeneous differential equation. This solution specifically accounts for the non-zero right-hand side, which in our problem is \( \tan(2t) - \frac{1}{2} e^t \).

To find a particular solution \( y_p \), one must craft a potential solution form that mirrors the form of the non-homogeneous part. For example, since our \( f(t) \) includes \( \tan(2t) \) and \( e^t \), we try a solution of the form \( y_p = A \tan(2t) + Be^{t} \).
The coefficients \( A \) and \( B \) are determined by substituting \( y_p \) into the original equation and solving for these unknowns. This ensures that \( y_p \) satisfies the equation when plugged back into the original differential equation.

The characteristic equation emerges from the homogeneous part of the differential equation. Getting to this equation involves setting \( f(t) = 0 \) in \( \frac{1}{2} y'' + 2y = 0 \), leading us to focus on the terms with \( y'' \) and \( y \).

We assume a solution of the form \( y = e^{rt} \), where \( r \) is a constant. Substituting this assumption into the homogeneous equation simplifies the problem to an algebraic equation involving \( r \).
This is known as the characteristic equation and it provides roots, \( r \), which indicate the nature of the solutions.

• If roots are real and distinct, solutions are exponential functions of form \( c_1 e^{r_1 t} + c_2 e^{r_2 t} \).
• If roots are real and equal, solutions include terms like \( c_1 e^{r_1 t} + c_2 t e^{r_1 t} \).
• Complex roots lead to sinusoidal solutions involving sine and cosine functions combined with exponentials.

The general solution to a non-homogeneous differential equation is a combination of both the homogeneous and particular solutions. It represents all possible solutions to the non-homogeneous equation.

For our specific problem, we have identified both a homogeneous solution \( y_h \) and a particular solution \( y_p \). By adding them together, we achieve the general solution:

• \( y = y_h(t) + y_p(t) \)
• This combination demonstrates how different parts of the differential equation are satisfied by either the homogeneous or particular solution components.

The general solution is beneficial because it offers the complete set of solutions satisfying both the differential equation and its accompanying initial or boundary conditions.