This chapter is about separable equations, namely, first-order differential equations \(y' = f(x,y)\) that can be rewritten in the form \(\displaystyle \frac {dy}{dx} = g(x)h(y). \qquad \hbox{(1)} \) The method of separation of variables is introduced in this chapter by playing around and tinkering with simple examples of (1) and “discovering” this method, after which it is completely justified using an argument involving the indefinite integrals of \(g(x)\) and \(1/h(y)\) and the chain rule. The distinction between explicit and implicit solutions of (1) is discussed and illustrated with many examples, some of which are accompanied by graphs of several solutions. Since there is always the possibility of making mistakes in algebra or integration, readers are encouraged to always check the functions they allege to be solutions by verifying that these functions truly satisfy the original form of Eq. (1). It is also shown how to check implicit solutions using the method of implicit differentiation. Intervals of existence of solutions are also examined. Even though most initial value problems involving equations of the form (1) can be solved using indefinite integration, a couple of examples suggest that it may be more convenient or even necessary to use definite integration since there are initial value problems, such as \(\displaystyle \frac {dy}{dx} = \frac {e^{-x^2}}{2y}, \quad y(1) = -2, \qquad \hbox{(2)} \) whose solution must be expressed in terms of a definite integral. Such is the case with (2) since the function exp \((-x^2)\) does not have an elementary antiderivative. Applications involving separable equations considered in this chapter include using Newton’s law of cooling to predict temperatures, such as a cup of hot coffee left on a table, and a derivation of the Cobb-Douglas production function, which is a formula used by economists to model the total production of goods in an economic system. The concluding section is about homogeneous first-order equations, which are equations that can be written in the form \(\displaystyle \frac {dy}{dx} = F\left (\frac {y}{x}\right ), \qquad \hbox{(3)} \) where F denotes a function of the ratio \(y/x\) . An equation that looks like (3) can be solved by replacing \(y/x\) with a new dependent variable, say z, which transforms it into a separable equation. A convenient test to ascertain whether a differential equation \(y' = f(x,y)\) is homogeneous without having to expend time and effort rewriting it as (3) is also discussed.

Separable and Related Equations

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发表日期：

2026年

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Leigh C. Becker

摘要

This chapter is about separable equations, namely, first-order differential equations \(y' = f(x,y)\) that can be rewritten in the form \(\displaystyle \frac {dy}{dx} = g(x)h(y). \qquad \hbox{(1)} \) The method of separation of variables is introduced in this chapter by playing around and tinkering with simple examples of (1) and “discovering” this method, after which it is completely justified using an argument involving the indefinite integrals of \(g(x)\) and \(1/h(y)\) and the chain rule. The distinction between explicit and implicit solutions of (1) is discussed and illustrated with many examples, some of which are accompanied by graphs of several solutions. Since there is always the possibility of making mistakes in algebra or integration, readers are encouraged to always check the functions they allege to be solutions by verifying that these functions truly satisfy the original form of Eq. (1). It is also shown how to check implicit solutions using the method of implicit differentiation. Intervals of existence of solutions are also examined. Even though most initial value problems involving equations of the form (1) can be solved using indefinite integration, a couple of examples suggest that it may be more convenient or even necessary to use definite integration since there are initial value problems, such as \(\displaystyle \frac {dy}{dx} = \frac {e^{-x^2}}{2y}, \quad y(1) = -2, \qquad \hbox{(2)} \) whose solution must be expressed in terms of a definite integral. Such is the case with (2) since the function exp \((-x^2)\) does not have an elementary antiderivative. Applications involving separable equations considered in this chapter include using Newton’s law of cooling to predict temperatures, such as a cup of hot coffee left on a table, and a derivation of the Cobb-Douglas production function, which is a formula used by economists to model the total production of goods in an economic system. The concluding section is about homogeneous first-order equations, which are equations that can be written in the form \(\displaystyle \frac {dy}{dx} = F\left (\frac {y}{x}\right ), \qquad \hbox{(3)} \) where F denotes a function of the ratio \(y/x\) . An equation that looks like (3) can be solved by replacing \(y/x\) with a new dependent variable, say z, which transforms it into a separable equation. A convenient test to ascertain whether a differential equation \(y' = f(x,y)\) is homogeneous without having to expend time and effort rewriting it as (3) is also discussed.