This chapter explores how to find solutions of second-order linear differential equations with constant coefficients, namely equations of the form \(\displaystyle a\frac {d^2x}{dt^2} + b\frac {dx}{dt} + cx = g(t), \qquad \hbox{(1)} \) where the constant \(a \ne 0\) and g represents a given function. The chapter begins with a description of the physical setup of a body sitting on a virtually frictionless air track and attached to a so-called ideal spring whose mass is negligible compared to that of the body. In one scenario, it is assumed that the only forces acting on the body are the restoring force of the spring and a damping force that is proportional to the velocity of the body. Newton’s second law and Hooke’s law are used to derive a differential equation that models the position x of the body at time t when it is displaced from its resting position and then released. This equation is of the form \(\displaystyle ax'' + bx' + cx = 0, \qquad \hbox{(2)} \) which is (1) with \(g(t) \equiv 0\) . This is the launching point for explaining how to find a general solution of a given homogeneous equation (2) when the associated characteristic equation \(\displaystyle a\lambda ^2 + b\lambda + c = 0 \qquad \hbox{(3)} \) has distinct real roots or a repeated real root. The case of complex roots begins with the simple equation \(\displaystyle x'' + {\omega }^{2}x = 0 \quad (\omega > 0). \qquad \hbox{(4)} \) The reason for using this particular equation is that it models the periodic motion of a body attached to an ideal spring when friction and other damping forces are totally absent and because the roots of the associated characteristic equation are \({\pm }i\omega \) . This suggests solutions of (4) are linear combinations of \(\sin {\omega {t}}\) and \(\cos {\omega {t}}\) and that there is a connection between complex characteristic roots and the sine and cosine functions. After this is shown to be the case, attention turns to finding a general solution of a given equation (2) when the characteristic equation for it has complex roots with nonzero real parts. Details on how all of this unfolds are presented in Sect. 10.3. Section 10.4 in troduces Euler’s formula, which defines the complex exponential function \(e^{it}\) . Instead of the usual definition involving the Maclaurin series for \(e^{t}\) and extending it to complex numbers, Euler’s formula is justified by explaining how to define \(e^{it}\) so that it yields the solutions \(\cos {t}\) and \(\sin {t}\) of two initial value problems involving (4) with \(\omega = 1\) . Among the other topics covered in this chapter are Cauchy-Euler equations in Sect. 10.5; linear homogeneous equations of higher order in Sect. 10.6; and an introduction to second-order nonhomogeneous linear equations in Sect. 10.7, that is, equations of the form (1) where \(g(t)\) is not identically zero. Sections 10.8 and 10.9 cover the methods of undetermined coefficients and variation of parameters.

Second-Order Linear Equations

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2026年

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Leigh C. Becker

摘要

This chapter explores how to find solutions of second-order linear differential equations with constant coefficients, namely equations of the form \(\displaystyle a\frac {d^2x}{dt^2} + b\frac {dx}{dt} + cx = g(t), \qquad \hbox{(1)} \) where the constant \(a \ne 0\) and g represents a given function. The chapter begins with a description of the physical setup of a body sitting on a virtually frictionless air track and attached to a so-called ideal spring whose mass is negligible compared to that of the body. In one scenario, it is assumed that the only forces acting on the body are the restoring force of the spring and a damping force that is proportional to the velocity of the body. Newton’s second law and Hooke’s law are used to derive a differential equation that models the position x of the body at time t when it is displaced from its resting position and then released. This equation is of the form \(\displaystyle ax'' + bx' + cx = 0, \qquad \hbox{(2)} \) which is (1) with \(g(t) \equiv 0\) . This is the launching point for explaining how to find a general solution of a given homogeneous equation (2) when the associated characteristic equation \(\displaystyle a\lambda ^2 + b\lambda + c = 0 \qquad \hbox{(3)} \) has distinct real roots or a repeated real root. The case of complex roots begins with the simple equation \(\displaystyle x'' + {\omega }^{2}x = 0 \quad (\omega > 0). \qquad \hbox{(4)} \) The reason for using this particular equation is that it models the periodic motion of a body attached to an ideal spring when friction and other damping forces are totally absent and because the roots of the associated characteristic equation are \({\pm }i\omega \) . This suggests solutions of (4) are linear combinations of \(\sin {\omega {t}}\) and \(\cos {\omega {t}}\) and that there is a connection between complex characteristic roots and the sine and cosine functions. After this is shown to be the case, attention turns to finding a general solution of a given equation (2) when the characteristic equation for it has complex roots with nonzero real parts. Details on how all of this unfolds are presented in Sect. 10.3. Section 10.4 in troduces Euler’s formula, which defines the complex exponential function \(e^{it}\) . Instead of the usual definition involving the Maclaurin series for \(e^{t}\) and extending it to complex numbers, Euler’s formula is justified by explaining how to define \(e^{it}\) so that it yields the solutions \(\cos {t}\) and \(\sin {t}\) of two initial value problems involving (4) with \(\omega = 1\) . Among the other topics covered in this chapter are Cauchy-Euler equations in Sect. 10.5; linear homogeneous equations of higher order in Sect. 10.6; and an introduction to second-order nonhomogeneous linear equations in Sect. 10.7, that is, equations of the form (1) where \(g(t)\) is not identically zero. Sections 10.8 and 10.9 cover the methods of undetermined coefficients and variation of parameters.