91Ó°ÊÓ

When solving a second-order linear differential equation with constant coefficients, finding the characteristic equation is a crucial step. This equation is derived from the form:

• \( ay^{\prime\prime} + by^{\prime} + cy = 0 \)

In this equation, \( a \), \( b \), and \( c \) are constants, and the characteristic equation will be:

• \( ar^2 + br + c = 0 \)

The characteristic equation is a key tool because its roots help us determine the behavior of the solution to the differential equation. For example, for the given problem, the characteristic equation is:

• \( r^2 - 6r + 9 = 0 \)

This equation will reveal whether the roots are real and distinct, repeated, or complex, guiding us to the correct form of the general solution.

In the study of characteristic equations, a repeated root occurs when the discriminant (\( b^2 - 4ac \)) is zero. This happens in our example with the characteristic equation \( r^2 - 6r + 9 = 0 \). Calculating:

• Discriminant = \( (-6)^2 - 4 \times 1 \times 9 = 0 \)

A zero discriminant means we have a double root. Double or repeated roots necessitate a particular form for the general solution. If our equation \( r = 3 \), this root repeats itself once.
When roots are repeated, the solution includes terms like:

• \( y(t) = (C_1 + C_2 t) e^{rt} \)

The additional \( C_2 t \) term differentiates the repeated root solution from solutions with distinct roots, ensuring that we generate a complete set of solutions.

The general solution to a second-order linear differential equation results from the form of the characteristic equation’s roots. In this example, because the root was repeated, our general solution follows this pattern:

• \( y(t) = (C_1 + C_2 t) e^{3t} \)

Here, \( C_1 \) and \( C_2 \) are constants determined by initial conditions; they represent arbitrary constants necessary to cover all possible solutions. The importance of general solutions lies in their ability to model real-life phenomena across different contexts, as they provide a comprehensive set of functions fitting the differential equation.
The exponential part, \( e^{3t} \), results from the repeated root \( r = 3 \), while the term \( C_1 + C_2 t \) covers the additional dimension of the solution space.

A second-order differential equation is identified by the presence of the second derivative, as in the equation:

• \( y^{\prime\prime} - 6y^{\prime} + 9y = 0 \)

They are linear when no terms depend on powers or products of \( y \) or its derivatives, aside from constant coefficients. These equations are significant in modeling systems with acceleration components, such as mechanical vibrations or oscillations.
Key steps in solving such an equation include:

• Identifying and forming the characteristic equation
• Determining the type of roots (real, repeated, or complex)
• Using roots to build the general solution

Second-order equations with constant coefficients are particularly manageable because their systematic solution steps—like those above—make them more approachable for students.