91Ó°ÊÓ

A differential equation can be classified by its order and whether it is linear or nonlinear. In this exercise, we are dealing with a **second-order homogeneous linear differential equation**. This type of equation has a standard form:
\[ a y^{\text{''}} + b y^{\text{'}} + c y = 0 \]
Here, the coefficients 'a', 'b', and 'c' are constants. The word 'homogeneous' indicates that there are no terms independent of the function 'y' on the left side of the equation. In our specific case, we have:
\[ y^{\text{''}} - 3y = 0 \]
The key identifying feature is that each term is a derivative of the function 'y' multiplied by a coefficient. Understanding this helps identify suitable solving methods.

To solve a homogeneous linear differential equation, we turn it into its **characteristic equation**. For our differential equation:
\[ y^{\text{''}} - 3y = 0 \]
This is achieved by assuming a solution of the form:
\[ y = e^{rt} \]
where 'r' is a constant to be determined. Substituting this assumed solution into the differential equation yields:
\[ (r^2 e^{rt}) - 3e^{rt} = 0 \]
By factoring out the common term \( e^{rt} \):
\[ e^{rt} (r^2 - 3) = 0 \]
Since \( e^{rt} \) is never zero, we are left with:
\[ r^2 - 3 = 0 \]
This is our characteristic equation. Solving this quadratic equation gives the roots, which are vital for forming the general solution.

After deriving the characteristic equation:
\[ r^2 - 3 = 0 \]
we solve for the roots to find:
\[ r = \text{±}\text{√}{3} \]
These roots lead us to the **general solution** of the differential equation. The general solution combines terms based on these roots:
\[ y(t) = C_1 e^{\text{√}{3}t} + C_2 e^{-\text{√}{3}t} \]
Here:

• \( C_1 \) and \( C_2 \) are arbitrary constants that will be determined by initial conditions or boundary values.
• Our solution includes both \( e^{\text{√}{3}t} \) and \( e^{-\text{√}{3}t} \), reflecting the two distinct roots of the characteristic equation.

By studying these aspects, we better understand both the structure and solutions of second-order homogeneous linear differential equations.