Book chapter
Linear Second Order Differential Equations with Constant Coefficients
Differential Equations, pp 81-180
2018
Abstract
This chapter is devoted to the detailed study of second order differential equations with constant coefficients. The methodology based on the roots of the characteristic equation has been the primary focus for the determination of the solution. Unlike the conventional pedagogic formats where other approaches for obtaining the solutions are presented separately either in the Appendix or in other chapters, all the methods to obtain the solutions are presented together. These methods include breaking the second order differential equation into a pair of coupled first order differential equations, Laplace transforms, as well as numerical methods based on Runge-Kutta using Matlab. This allows the reader to compare the results from different approaches. The results also include detailed analysis of the phase portraits which help explain the stability of the systems modeled through these differential equations. While homogeneous differential equations are solved through the multiple ways described above, non-homogeneous differential equations are solved through the use of method of variation of parameters as well as the method of undetermined coefficients in order to obtain the particular solution. The results from these two approaches for obtaining the particular solutions are compared and explanations are provided appropriately in cases where the particular solutions appear different. Even in the case of the non-homogeneous differential equations, Laplace transforms as well as ODE based methods are employed as additional verification steps. The examples cover a substantial number of different cases and the solution in each case is prepared to be self-contained with appropriate theory.
Second order differential equations occur in the study of models such as those that describe the behavior of mechanical and electrical chemical transportation systems. For example, the best way to describe the motion of a pendulum, the characteristics of the current or voltage in electric circuits and or the flow of chemicals is through the use of second order differential equations. If the non-zero roots of the characteristic equation are real and positive, the system is unstable. The system is asymptotically stable when the roots are a conjugate pair with the real part being negative. The complete solution to a non-homogeneous equation can be obtained from the homogeneous and particular solutions. If initial conditions are given, solution can also be obtained using Laplace transforms and Runge-Kutta methods.
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Details
- Title
- Linear Second Order Differential Equations with Constant Coefficients
- Creators
- P. Mohana Shankar - Drexel University
- Publication Details
- Differential Equations, pp 81-180
- Publisher
- CRC Press
- Edition
- 1
- Resource Type
- Book chapter
- Language
- English
- Academic Unit
- Electrical and Computer Engineering
- Other Identifier
- 991019186962004721