The whole course in one sitting. Classify an equation first, because the classification picks the method: separable, linear or exact for first order; the characteristic equation for constant coefficients; a guess, variation of parameters or the Laplace transform for forcing; eigenvalues for a linear system and the Jacobian for a nonlinear one; a series when the coefficients vary; and a numerical method when nothing else applies. Then read what the answer says, and check it against the direction field or the phase portrait.

A toolbox at the end of an apprenticeship: every tool is familiar, and what is tested is picking the right one for the job in front of you, quickly.

Ask three questions of every equation. What is it — its order, linear or not, constant or varying coefficients, forced or not, one equation or a system? What is asked — a formula, a value, a long-run behaviour, a verdict on stability? And what does the theory promise — existence and uniqueness, the dimension of the solution space, the reach of a series? The first answer picks the method; the other two say what to compute and what to trust.

• First order: separable, $y'=g(x)h(y)$; linear, with the integrating factor $\mu=e^{\int p\,dx}$; exact, with a potential $F$ such that $F_{x}=M$ and $F_{y}=N$.
• Constant coefficients: the characteristic equation; forcing by undetermined coefficients, variation of parameters or the Laplace transform.
• Systems: the eigenvalues of $A$, read from the trace and the determinant; a nonlinear system through the Jacobian at each equilibrium.
• Varying coefficients: a power series about an ordinary point, a Frobenius series at a regular singular point.
• Anything else, or a check on all of the above: Euler, Heun or Runge–Kutta, with the step chosen by halving.