Differential Equations and Linear Algebra, 4th Edition

1.6 Substitution Methods and Exact Equations

The first-order differential equations we have solved in the previous sections have all been either separable or linear. But many applications involve differential equations that are neither separable nor linear. In this section we illustrate (mainly with examples) substitution methods that sometimes can be used to transform a given differential equation into one that we already know how to solve.

For instance, the differential equation

$${\displaystyle \frac{dy}{dx}}=f(x,y),$$ (1)

with dependent variable y and independent variable x, may contain a conspicuous combination

$$v=\alpha (x,y)$$ (2)

of x and y that suggests itself as a new independent variable v. Thus the differential equation

$${\displaystyle \frac{dy}{dx}}={(x+y+3)}^2$$

practically demands the substitution ...