Differential Equations and Linear Algebra, 4th Edition

1.3 Slope Fields and Solution Curves

Consider a differential equation of the form

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

where the right-hand function $f(x,y)$ involves both the independent variable x and the dependent variable y. We might think of integrating both sides in (1) with respect to x, and hence write $y(x)=\int f(x,y(x))dx+C$. However, this approach does not lead to a solution of the differential equation, because the indicated integral involves the unknown function y(x) itself, and therefore cannot be evaluated explicitly. Actually, there exists no straightforward procedure by which a general differential equation can be solved explicitly. Indeed, the solutions ...