91Ó°ÊÓ

The characteristic equation is a crucial step in solving linear differential equations, especially those with constant coefficients. In our case, the differential equation is given by:\[ y'' + 2y' - 8y = 0 \]First, we turn this into the characteristic equation by substituting:

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

This results in a quadratic equation:\[ r^2 + 2r - 8 = 0 \]Quadratic equations like this are common and help us find the values of \( r \), which are essential for determining the solution of the differential equation.
Quadratic equations in this context often emerge from the relationship between exponential functions and differential equations, where each root represents an exponential solution component of the differential equation.

Once we have our characteristic equation solved, we can write the general solution for the differential equation. The general solution is formed based on the identified roots of the characteristic equation.
In this scenario, we use the distinct real roots obtained from the characteristic equation to express the general solution. The solution takes the form:\[ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} \]where:

• \( C_1 \) and \( C_2 \) are arbitrary constants.
• \( r_1 \) and \( r_2 \) are the roots obtained from the characteristic equation.

For our differential equation, the roots \( r_1 = -4 \) and \( r_2 = 2 \) yield the general solution:\[ y(x) = C_1 e^{-4x} + C_2 e^{2x} \]This form is a combination of exponential functions that satisfy the original differential equation.
At its core, the general solution represents a family of all possible solutions, adjustable by altering the values of \( C_1 \) and \( C_2 \).

Distinct real roots in the characteristic equation signify that the quadratic equation \( r^2 + 2r - 8 = 0 \) results in two different values for \( r \). This is often the most straightforward case to handle in differential equations.
Solving the quadratic yields the roots:

• \( r_1 = -4 \)
• \( r_2 = 2 \)

These roots provide important information about the behavior of the differential equation's solution.
If the roots are distinct and real, each exponential part \( e^{r_1 x} \) and \( e^{r_2 x} \) contributes uniquely to the general solution. The exponential terms describe how the solution behaves over time, typically exhibiting growth or decay
based on the sign of the roots.
Distinct real roots suggest that no additional multiplicative factors, like \( x \), are needed in the solutions, which would happen in cases of repeated roots.