ABSTRACT
In the previous chapters, it was seen how certain types of first-order differential equations --- directly integrable, separable and linear --- can be identified and solved. However, many first-order equations do not fall into one of these classes. When this happens, a clever “substitution” might be useful. This refers to replacing the unknown function, say, y, in the differential equation with some formula of the dependent variable and another unknown function, say, u, to obtain a new differential equation that is either directly integrable, separable or linear. The original differential equation is then solved by combining the solution of the new differential equation with the formula relating the originally unknown functions u and y.
This chapter begins with a general discussion of substitution as just described, both to obtain explict solutions and implicit solutions. The rest of the chapter deals in detail with identifying three classes of first-order differential equations in which particular substitutions are likely to work, and to the details of using those substitutions. These classes include those equations for which “linear substitutions” are appropriate, “homogenous” equations and Bernoulli equations.
