Characteristic Equation

Definition

The characteristic equation is a fundamental concept in the study of second-order linear differential equations. It is an algebraic equation that is derived from the differential equation and is used to determine the general solution of the differential equation.

5 Must Know Facts For Your Next Test

1. The characteristic equation is formed by substituting the trial solution $y = e^{rt}$ into the homogeneous second-order linear differential equation.
2. The roots of the characteristic equation determine the behavior of the solutions to the differential equation, whether they are real, complex, or repeated.
3. If the roots of the characteristic equation are real and distinct, the general solution to the differential equation will be a linear combination of two exponential functions.
4. If the roots of the characteristic equation are complex conjugates, the general solution will be a linear combination of sinusoidal and exponential functions.
5. If the roots of the characteristic equation are repeated, the general solution will involve a linear combination of exponential functions and polynomials.

Review Questions

• Explain how the characteristic equation is derived from a second-order linear homogeneous differential equation.
  • To derive the characteristic equation, we start with a second-order linear homogeneous differential equation in the form $a(t)y'' + b(t)y' + c(t)y = 0$. We then substitute a trial solution of the form $y = e^{rt}$ into the equation, which results in an algebraic equation in the variable $r$. This algebraic equation is the characteristic equation, and its roots determine the form of the general solution to the differential equation.
• Describe the relationship between the roots of the characteristic equation and the behavior of the solutions to the second-order linear homogeneous differential equation.
  • The roots of the characteristic equation play a crucial role in determining the behavior of the solutions to the second-order linear homogeneous differential equation. If the roots are real and distinct, the solutions will be a linear combination of two exponential functions. If the roots are complex conjugates, the solutions will be a linear combination of sinusoidal and exponential functions. If the roots are repeated, the solutions will involve a linear combination of exponential functions and polynomials. The nature of the roots, whether real, complex, or repeated, directly influences the form of the general solution to the differential equation.
• Analyze how the characteristic equation can be used to determine the general solution of a second-order linear homogeneous differential equation.
  • The characteristic equation is the key to determining the general solution of a second-order linear homogeneous differential equation. By solving the characteristic equation and finding its roots, we can identify the appropriate form of the general solution. If the roots are real and distinct, the general solution will be a linear combination of two exponential functions. If the roots are complex conjugates, the general solution will be a linear combination of sinusoidal and exponential functions. If the roots are repeated, the general solution will involve a linear combination of exponential functions and polynomials. The characteristic equation, therefore, provides the essential information needed to construct the general solution to the differential equation.

Related terms

Homogeneous Differential Equation: A second-order linear differential equation is said to be homogeneous if the right-hand side of the equation is zero, meaning there is no forcing function or input term.

Roots of the Characteristic Equation: The roots of the characteristic equation are the values of the variable that satisfy the equation and determine the form of the general solution to the differential equation.

Auxiliary Equation: The auxiliary equation is another name for the characteristic equation, as it is used to find the auxiliary functions that form the general solution of the differential equation.