91Ó°ÊÓ

A characteristic equation is a crucial step when dealing with homogeneous linear differential equations. It stems from the original differential equation by substituting the derivatives with powers of a variable, usually denoted by \(r\). For a second-order differential equation such as \(y'' - y' - 11y = 0\), each derivative is replaced as follows:

• \(y''\) becomes \(r^2\)
• \(y'\) becomes \(r\)
• \(y\) becomes 1

This transformation results in a quadratic equation, known as the characteristic equation. For the given differential equation, the characteristic equation is \(r^2 - r - 11 = 0\). Solving this gives us the roots of the equation, which play a key role in determining the general solution of the differential equation.

The general solution of a second-order linear differential equation is constructed from the roots of its characteristic equation. These roots are used to form exponential solutions of the differential equation. For each distinct root \(r_i\), an exponential term \(e^{r_ix}\) is part of the solution.
In our exercise, the characteristic equation \(r^2 - r - 11 = 0\) provides two distinct roots: \(r = 4\) and \(r = -3\). Consequently, the general solution consists of two exponential functions combined.

• \(y_1 = e^{4x}\)
• \(y_2 = e^{-3x}\)

Thus, the general solution is the linear combination of these, expressed as \(y = c_1e^{4x} + c_2e^{-3x}\), where \(c_1\) and \(c_2\) are constants determined by initial or boundary conditions.

Homogeneous linear differential equations have a standard form where all terms are dependent on the function and its derivatives, equaling zero, such as \(y'' - y' - 11y = 0\). These equations are important because their solutions exhibit properties of linear combinations, allowing the superposition of solutions which means we can add multiple solutions together to form new solutions.
The process of solving these equations involves finding the characteristic equation, which reveals the nature of the solutions. If the characteristic equation yields distinct real roots, as in our exercise, the solutions are exponentials based on those roots.

• This results in solutions that are linear combinations of exponential terms.
• In cases where roots are complex or repeated, the form of the general solution changes accordingly.

Using the properties of superposition and linearity, solutions to homogeneous linear differential equations are versatile and can represent a wide range of behaviors.