Abstract
In this chapter we study second-order linear differential equations of the form
and their applications to classical mechanics and electrical circuits. These applications are standard fare and a centerpiece in both elementary physics and engineering courses, and they serve as prototypes for oscillating systems, oscillating systems with dissipation, or damping, and forced vibrations that occur in all areas of pure and applied science. In the final sections of the chapter we extend the coverage to linear equations with variable coefficients.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Notation: Any complex number \(z\) can be written \(z=u+iv\), where \(u\) and \(v\) are real numbers; \(u\) is called the real part of \(z\) and \(v\) is called the imaginary part of \(z\). Similarly, if \(z(t)=u(t)+iv(t)\) is a complex function, then \(u(t)\) and \(v(t)\) are its real and imaginary parts, respectively. The numbers \(u+iv\) and \(u-iv\) are called complex conjugates.
- 2.
If \(a+bi=0\), where \(A\) and \(b\) are real, then, necessarily, \(a=b=0\).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishers, Switzerland
About this chapter
Cite this chapter
Logan, J. (2015). Second-Order Linear Equations. In: A First Course in Differential Equations. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-17852-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-17852-3_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-17851-6
Online ISBN: 978-3-319-17852-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)