Repeated roots

Definition

Repeated roots occur when a polynomial equation has a root that appears more than once. In the context of second-order linear differential equations, repeated roots influence the general solution, affecting the form and nature of the solutions derived from the characteristic equation associated with the differential equation.

5 Must Know Facts For Your Next Test

1. When a second-order linear differential equation has repeated roots, the general solution includes a logarithmic term alongside exponential terms.
2. The repeated root scenario arises when the discriminant of the characteristic polynomial is zero.
3. In terms of function forms, if the repeated root is $r$, the solutions take the form $c_1 e^{rt} + c_2 t e^{rt}$, where $c_1$ and $c_2$ are constants.
4. Repeated roots suggest that the solutions converge towards a single behavior rather than diverging, showing their dependence on each other.
5. Understanding repeated roots helps in predicting system behaviors in applications like mechanical vibrations and electrical circuits.

Review Questions

• How does the presence of repeated roots in a characteristic equation affect the general solution of a second-order linear differential equation?
  • The presence of repeated roots in a characteristic equation directly impacts the form of the general solution. When a second-order linear differential equation has repeated roots, the solution must include both an exponential term and a logarithmic term. Specifically, if $r$ is the repeated root, the general solution is given by $c_1 e^{rt} + c_2 t e^{rt}$, which indicates that both solutions share a common exponential behavior but are adjusted by a factor of time $t$.
• Discuss how identifying repeated roots helps in solving homogeneous second-order linear differential equations more effectively.
  • Identifying repeated roots allows for a more accurate formulation of the general solution to homogeneous second-order linear differential equations. By recognizing that two roots are identical, it’s possible to apply specific techniques to construct solutions that reflect this situation. This awareness leads to including additional terms in the solution process, ensuring that all potential behaviors of the system are captured rather than missing critical components that arise from repeated roots.
• Evaluate the implications of repeated roots in physical systems modeled by second-order linear differential equations, particularly regarding stability and oscillations.
  • Repeated roots in physical systems modeled by second-order linear differential equations often indicate a system's tendency toward critical damping or an equilibrium point without oscillation. This means that rather than oscillating around an equilibrium position, the system will return to stability in a smooth manner. Analyzing these scenarios provides insight into system stability; for example, in mechanical or electrical systems, this behavior could signal an ideal state without overshooting or sustained oscillations, crucial for designing stable control systems.

Related terms

Characteristic equation: An equation obtained from a differential equation by substituting solutions of the form $e^{rt}$, leading to a polynomial whose roots help determine the behavior of the solutions.

Homogeneous equation: A differential equation where all terms are dependent on the function and its derivatives, typically leading to a characteristic equation whose roots are analyzed for solution forms.

Fundamental solutions: The set of linearly independent solutions of a homogeneous differential equation, which can be combined to form the general solution.