91Ó°ÊÓ

The characteristic equation is a key concept when solving linear differential equations with constant coefficients. It arises because of the method we use to determine the possible solutions of the differential equation. The idea is to convert the original differential equation into an algebraic equation, known as the characteristic equation, where the derivatives are replaced by powers of a variable, usually denoted as \( r \).

In the given differential equation:

• We assumed a solution of the form \( y = e^{rt} \)
• We substituted into the differential equation to get an equation in terms of \( r \)
• This yielded the characteristic equation: \( r^4 - 13r^2 + 36 = 0 \)

This characteristic equation tells us the values of \( r \) for which our assumed solution form \( e^{rt} \) is a solution of the differential equation.

The general solution of a differential equation is essential because it encompasses all possible solutions, given the linearity and rules of the differential equation. For linear differential equations with constant coefficients, once we have the characteristic equation:

• We solve it to find the roots.
• The nature of these roots (real and distinct, real and repeated, or complex) dictates the form of the general solution.

In this case, after solving the characteristic equation, we obtained distinct real roots \( r = 3, -3, 2, -2 \). These roots correspond to exponential solutions:\[y(t) = C_1 e^{3t} + C_2 e^{-3t} + C_3 e^{2t} + C_4 e^{-2t}\]where each \( C_i \) is an arbitrary constant. This formula represents the complete general solution for our differential equation.

Real roots in the context of the characteristic equation mean the solutions are exponential in nature. They tell us about the behavior and type of the function solutions. To determine these roots:

• We solved the characteristic equation like a polynomial.
• Substituted \( x = r^2 \), transforming the equation into a simple quadratic: \( x^2 - 13x + 36 = 0 \).
• Using the quadratic formula, we found \( x = 9 \) and \( x = 4 \).
• This gave us the roots \( r = \pm 3 \) and \( r = \pm 2 \), since \( r^2 = x \).

Distinct real roots lead to distinct terms in the general solution, providing a separate exponential term for each root.

Linear differential equations are equations involving a function and its derivatives with linear combinations. In such equations, each term is either a constant or the product of a constant and a function of the independent variable, without any multiplications between derivatives of the function.

The equation in the exercise, \( y^{(4)} + 36y = 13y'' \), is a linear differential equation. Here:

• The highest order derivative is \( y^{(4)} \).
• It includes simple linear terms of \( y \).

Because it is linear with constant coefficients, we could employ the technique of utilizing the characteristic equation to find the general solution efficiently. This property allows for systematic approaches to be applied, making it easier to solve compared to nonlinear differential equations.