91Ó°ÊÓ

The characteristic equation is a fundamental tool used in solving linear differential equations. In the case of a third-order differential equation like the one given: \[ y^{\prime \prime \prime} - 4y^{\prime \prime} - 11y^{\prime} + 30y = 0, \]we need to rewrite it in terms of an exponential function substitution. By assuming a solution of the form \( y = e^{rx} \), we can replace the derivatives in the equation by powers of \( r \), ultimately arriving at the characteristic polynomial:\[ r^3 - 4r^2 - 11r + 30 = 0. \]This polynomial equation is called the characteristic equation. Solving it gives us the values of \( r \) that characterize the behavior of solutions to the differential equation.

When solving the characteristic equation \( r^3 - 4r^2 - 11r + 30 = 0 \), we found its roots to be \( r = 1, r = 3, \) and \( r = -10 \). These roots are distinct, meaning they are unique and do not repeat. Having distinct real roots provides us with a straightforward way to construct the general solution. Each distinct root corresponds to an exponential component in the solution of the differential equation. This results in a solution that is a simple linear combination of these exponential terms.

With the distinct real roots \( r = 1, r = 3, \) and \( r = -10 \), the general solution to the differential equation can be expressed as:\[ y(x) = C_1 e^x + C_2 e^{3x} + C_3 e^{-10x}, \]where \( C_1, C_2, \) and \( C_3 \) are arbitrary constants. These constants will be determined based on initial conditions or boundary values given in specific problems. Each term in this solution corresponds directly to one of the roots of the characteristic equation, highlighting how the distinct and real nature of the roots simplifies the form of the solution.

The Rational Root Theorem is a handy technique for finding the possible rational roots of a polynomial equation. It was crucial in solving our problem's characteristic equation \( r^3 - 4r^2 - 11r + 30 = 0 \). This theorem states that any rational root of a polynomial, with integer coefficients, is in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. In our example, the constant term is 30, and the leading coefficient is 1.

• Factors of 30: \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30 \)
• Factors of 1: \( \pm 1 \)

Using this theorem provided us a manageable list of potential rational roots that we could test, leading us efficiently to the roots \( r = 1, r = 3, \) and \( r = -10 \). Once these roots were identified, we could apply them directly in deriving the general solution.