This chapter begins with an introduction to integrating factors and how they are used to find solutions of the standard form \(\displaystyle \frac {dx}{dt} + p(t)x = q(t) \qquad \hbox{(1)} \) of the first-order linear equation \( a_1(t)x' + a_0 (t)x = b(t). \) Furthermore, an existence and uniqueness theorem for the initial value problem \(\displaystyle \frac {dx}{dt} + p(t)x = q(t), \quad x(t_0) = x_0 \qquad \hbox{(2)} \) is proved by employing an integrating factor. Gronwall’s inequality, a result that yields a bound for an unknown nonnegative continuous function f satisfying the inequality \(\displaystyle f(t) \le K + \left |\int _{t_0 }^t p(s)f(s)\,ds\right |, \) where \(K>0\) is a constant and p is a known nonnegative continuous function, is derived with the help of an integrating factor. Gronwall’s inequality is used to determine the extent to which two solutions of (1) can differ from each other over a given interval. It is instrumental in establishing one of the existence and uniqueness theorems for first-order equations in Chap. 9 . An alternative to finding the solution of the initial value problem (2) is the variation of parameters method. Its introduction here for first-order linear equations paves the way for explaining how to use this method to solve second-order nonhomogeneous linear equations (see Chap. 10 ). Furthermore, the variation of parameters formula \(\displaystyle x(t) = z(t,t_0)x_0 + \int _{t_0}^t z(t,s)q(s)\,ds \qquad \hbox{(3)} \) is derived, where \(z(t,s)=\exp \left (-\int _{s}^t p(u)\,du\right )\) . A matrix analog of (3) for nonhomogeneous linear systems is one of the topics in Chap. 13 . It is pointed out that solutions of differential equations as simple as (1) may have to be expressed in terms of functions that are defined in terms of definite integrals, such as Dawson’s integral and the sine integral function. The last two sections of this chapter deal with two nonlinear first-order differential equations and their solutions: the Bernoulli equation \(\displaystyle \frac {dx}{dt} + p(t)x = q(t)x^{n} \) and the Riccati equation \(\displaystyle \frac {dx}{dt} = a(t)x^2 + b(t)x + c(t), \) both of which are important in engineering, physics, and other fields.

First-Order Linear and Related Equations

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2026年

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Leigh C. Becker

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This chapter begins with an introduction to integrating factors and how they are used to find solutions of the standard form \(\displaystyle \frac {dx}{dt} + p(t)x = q(t) \qquad \hbox{(1)} \) of the first-order linear equation \( a_1(t)x' + a_0 (t)x = b(t). \) Furthermore, an existence and uniqueness theorem for the initial value problem \(\displaystyle \frac {dx}{dt} + p(t)x = q(t), \quad x(t_0) = x_0 \qquad \hbox{(2)} \) is proved by employing an integrating factor. Gronwall’s inequality, a result that yields a bound for an unknown nonnegative continuous function f satisfying the inequality \(\displaystyle f(t) \le K + \left |\int _{t_0 }^t p(s)f(s)\,ds\right |, \) where \(K>0\) is a constant and p is a known nonnegative continuous function, is derived with the help of an integrating factor. Gronwall’s inequality is used to determine the extent to which two solutions of (1) can differ from each other over a given interval. It is instrumental in establishing one of the existence and uniqueness theorems for first-order equations in Chap. 9 . An alternative to finding the solution of the initial value problem (2) is the variation of parameters method. Its introduction here for first-order linear equations paves the way for explaining how to use this method to solve second-order nonhomogeneous linear equations (see Chap. 10 ). Furthermore, the variation of parameters formula \(\displaystyle x(t) = z(t,t_0)x_0 + \int _{t_0}^t z(t,s)q(s)\,ds \qquad \hbox{(3)} \) is derived, where \(z(t,s)=\exp \left (-\int _{s}^t p(u)\,du\right )\) . A matrix analog of (3) for nonhomogeneous linear systems is one of the topics in Chap. 13 . It is pointed out that solutions of differential equations as simple as (1) may have to be expressed in terms of functions that are defined in terms of definite integrals, such as Dawson’s integral and the sine integral function. The last two sections of this chapter deal with two nonlinear first-order differential equations and their solutions: the Bernoulli equation \(\displaystyle \frac {dx}{dt} + p(t)x = q(t)x^{n} \) and the Riccati equation \(\displaystyle \frac {dx}{dt} = a(t)x^2 + b(t)x + c(t), \) both of which are important in engineering, physics, and other fields.