Choosing an ODE method

For a first- or second-order ODE:

1. Classify the order, linearity and domain restrictions.
2. If it is first order, test separability: can all $y$ terms be placed with $dy$ and all $x$ terms with $dx$?
3. If a first-order equation is linear, normalise it and use an integrating factor for $$y'+p(x)y=q(x).$$
4. If it has the form $M\,dx+N\,dy=0$, test exactness by comparing $M_y$ and $N_x$ on a suitable domain, then construct a potential when the test applies.
5. If a first-order equation has none of these structures, look for a substitution: $y/x$ suggests $y=xv$, while a power $y^n$ suggests a Bernoulli substitution. A substitution helps when the new derivative can be written using terms already present.
6. For a second-order equation, check whether it is linear with constant coefficients. If so, solve the characteristic equation and read the root type. For a forcing term that is polynomial, exponential or trigonometric, find one particular solution by undetermined coefficients, correcting the trial for resonance.
7. If no analytic method applies, use a direction field and a numerical scheme. Euler is the tangent-line method; Heun improves it by averaging a predicted pair of endpoint slopes.
8. Substitute the result back into the original equation and recheck the initial or boundary conditions.