91Ó°ÊÓ

A homogeneous differential equation is a special type of equation that has equal terms on both sides when set to zero. In our problem, the given differential equation is \( y'' + 9y = 0 \). This is referred to as a **second-order linear homogeneous differential equation** because:

• It's second-order, meaning it involves the second derivative \( y'' \).
• It's linear, as the equation includes no products or powers of the unknown function \( y \) or its derivatives other than the first power.
• It's homogeneous because all terms are dependent on \( y \) or its derivatives and the equation equals zero.

These equations are important as they appear frequently in physics and engineering, modeling many natural phenomena, like oscillations or waves. To solve them, we usually find solutions that satisfy given initial or boundary conditions.

The characteristic equation is a tool that helps us solve homogeneous differential equations, especially those with constant coefficients. When faced with the equation \( y'' + 9y = 0 \), we derive its characteristic equation as follows:
Write the equation as "characteristic equation" form: let \( y = e^{rx} \). Then, \( y' = re^{rx} \) and \( y'' = r^2e^{rx} \).
Substitute into the differential equation: \( r^2e^{rx} + 9e^{rx} = 0 \).
Factor the common term out: \( e^{rx}(r^2 + 9) = 0 \).
This leads to \( r^2 + 9 = 0 \), our characteristic equation.
By solving \( r^2 + 9 = 0 \), we find \( r = \pm 3i \).
These complex roots indicate that our solutions will involve trigonometric functions:

• \( C_1 \cos(3x) \)
• \( C_2 \sin(3x) \)

These form the general solution to a homogeneous differential equation.

Trigonometric functions play a crucial role in solving differential equations, especially when dealing with complex roots of the characteristic equation. In the boundary-value problem, our general solution involves \( C_1 \cos(3x) + C_2 \sin(3x) \). Here's why these functions are important:

• **Cosine and sine functions** are periodic, meaning they repeat values over regular intervals. This makes them ideal for modeling wave-like behavior in systems.
• The coefficients \( C_1 \) and \( C_2 \) are determined using initial or boundary conditions, such as \( y(0) = 4 \) and \( y\left(\frac{\pi}{3}\right) = -4 \).
• The angles within the functions (e.g., \( 3x \)) result from the complex roots \( \pm 3i \) derived from the characteristic equation.

By applying initial or boundary conditions, we adjust the trigonometric functions to fit specific problems, achieving a particular solution that satisfies given constraints.