91Ó°ÊÓ

The characteristic equation is a crucial concept in understanding how to solve linear differential equations, especially those with constant coefficients. It acts as a bridge between a differential equation and a simpler polynomial equation. This transformation helps greatly in finding solutions to the original differential equation.
To derive the characteristic equation from a linear homogeneous differential equation like \( ay'' + by' + cy = 0 \), we replace each derivative of \( y \) with a power of \( r \). Thus, \( y'' \) becomes \( r^2 \), \( y' \) becomes \( r \), and \( y \) becomes a constant. This creates a polynomial \( ar^2 + br + c = 0 \).
The solutions to this new polynomial equation, primarily the roots, hold important insights into the general solution of the differential equation.

• Each root corresponds to a component of the solution.
• Real and distinct roots yield simple exponential functions.
• Complex roots produce oscillating functions, involving sine and cosine.
• Repeated roots, as we will discuss, lead to slightly more complex solutions.

A homogeneous differential equation is a special type of differential equation where all terms involve the dependent variable or its derivatives, and there is no external forcing term. The equation given in the exercise, \( y'' + 4y' + 4y = 0 \), is a second-order linear homogeneous differential equation.
These equations are recognized by the form \( ay'' + by' + cy = 0 \), where the right-hand side is zero. The simplicity and structure of homogeneous equations make them easier to analyze and solve compared to non-homogeneous ones.

• Solving homogeneous equations involves finding the characteristic equation, a step we've already discussed.
• The solutions are typically expressed in terms of the exponential function \( e \), based on the nature of the roots of the characteristic equation.

The absence of non-zero terms on the right simplifies the analysis and solution of the differential equation, focusing strategies solely on properties of linearity and initial conditions.

Repeated roots in the context of solving linear differential equations indicate a specific solution pattern. When the characteristic equation has repeated roots, the general solution of the differential equation requires a modification from the standard forms.
The repeated roots formula is applied when you encounter a repeated root, say \( r = \alpha \). The solution instead involves terms of the following form:

• For a simple root \( r \), the solution is \( y(t) = C e^{rt} \).
• For a repeated root \( r = -2 \) as in the given problem, the general solution is \( y(t) = (C_1 + C_2 t) e^{-2t} \).

This formula accounts for the repeated nature by multiplying one of the exponential terms by the independent variable \( t \), introducing a linear adjustment. This approach ensures the solution remains valid and comprehensive, reflecting the complexity introduced by the repetition of roots. In our example, \( r = -2 \) appears twice, hence the term \( (C_1 + C_2 t) \) compensates for the repetition.