Section 9.2 Numerical Methods
Consider the case when the right hand side depends only on \(t\text{:}\)
\begin{equation*}
\dot x = f(t),\;x(t_0)=x_0.
\end{equation*}
From the Fundamental Theorem of Calculus we know that the solution of this equation is
\begin{equation*}
x(t) = x_0 + \displaystyle\int_{t_0}^t\!\! f(\tau)d\tau
\end{equation*}
Hence all methods we learned for numerical quadrature are automatically also numerical methods for solving these ODEs! In fact, all main methods to solve ODEs, even when the right hand side depends on the variable(s) \(x\text{,}\) are a direct generalization of numerical quadrature methods. For all of the few methods we will illustrate next, we will point out to which quadrature method corresponds.