7: First-Order ODEs
Reference: Boyce and DiPrima, Chapter 2
The general first-order differential equation for the function \(y=y(x)\) is written as
\[\frac{d y}{d x}=f(x, y), \nonumber \]
where \(f(x, y)\) can be any function of the independent variable \(x\) and the dependent variable \(y\) . We first show how to determine a numerical solution of this equation, and then learn techniques for solving analytically some special forms of Equation \ref{7.1}, namely, separable and linear first-order equations.