3: First-Order ODEs
The general first-order differential equation for the function \(y = y(x)\) is written as \[\label{eq:1}\frac{dy}{dx}=f(x,y),\] 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 \(\eqref{eq:1}\), namely, separable and linear first-order equations.