91Ó°ÊÓ

A characteristic equation is an essential part of solving linear differential equations. It helps us find the behavior of the solutions based on certain parameters. For an ordinary differential equation (ODE) with constant coefficients like \(y'' - 13y' + 42y = 0\), the characteristic equation is derived by replacing the derivatives with powers of \(r\) in the following manner:

• The second derivative \(y''\) becomes \(r^2\).
• The first derivative \(y'\) turns into \(r\).
• The term without a derivative keeps its constant \(42\).

So, the characteristic equation associated with the ODE is \(r^2 - 13r + 42 = 0\). The roots of this equation are critical, as they represent the exponents in the exponential solutions to the differential equation. By determining these roots, we can construct the solutions of the ODE.
This equation will either have:

• Real and distinct roots, leading to solutions in the form of distinct exponential functions.
• Complex roots, resulting in exponential solutions mixed with sine and cosine.
• Repeated roots, necessitating additional polynomial terms.

In this exercise, the roots are real and distinct, making the solution straightforward.

The quadratic formula is a universal tool used to find the roots of any quadratic equation. In cases where quadratic equations do not factor neatly, the quadratic formula comes in handy.
For our characteristic equation, \(r^2 - 13r + 42 = 0\), the formula is employed as follows:

• The general quadratic formula is: \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Comparing with our characteristic equation \(a = 1\), \(b = -13\), \(c = 42\).
• Substituting these into the formula, we solve \( r = \frac{13 \pm \sqrt{(-13)^2 - 4 \cdot 1 \cdot 42}}{2 \cdot 1} \).
• Simplifying further gives us \( r = \frac{13 \pm \sqrt{169 - 168}}{2} = \frac{13 \pm 1}{2}\).
• Thus, the roots are \( r_1 = 6 \) and \( r_2 = 7 \).

This method ensures the roots are found correctly even when they cannot be easily guessed or factored, reinforcing accuracy and efficiency.

In solving a linear differential equation, arriving at the general solution is a crucial step. This solution encapsulates all possible specific solutions of the differential equation. Once we've determined the roots of the characteristic equation, constructing the general solution becomes straightforward.

• If the roots \( r_1 \) and \( r_2 \) from the characteristic equation are distinct and real, as in \( 6 \) and \( 7 \), the general solution takes the form: \( y(x) = c_1 e^{r_1 x} + c_2 e^{r_2 x} \).
• For our example, substituting the roots results in: \( y(x) = c_1 e^{6x} + c_2 e^{7x} \).

The constants \( c_1 \) and \( c_2 \) are determined by the initial conditions of the specific problem, representing the contributions from each fundamental solution.
This form of the solution shows how the function \(y(x)\) will behave over time or space, based on the interaction of these exponential components. It offers a comprehensive view of all possible dynamics governed by the differential equation. Understanding the role of these parameters \( c_1 \) and \( c_2 \) is important for solving practical problems that rely on definite values.