principle of dominant balance
The principle of dominant balance is the practical art of guessing which terms in an equation actually matter in a given limit, by assuming that two of them are roughly the same size and dominate the rest. Faced with an equation crowded with terms of unknown magnitude, you do not solve it head-on; you hypothesise that a pair of terms balance each other while the others are negligible, work out what that implies, and then check the hypothesis for consistency. It is the first move in nearly every asymptotic calculation.
How it works: suppose an equation has several terms whose sizes you do not yet know. You pick two candidate terms, declare them to be of comparable magnitude (in balance), and provisionally drop all the others. Solving the simplified balance tells you the scale of the unknown, which you then use to estimate the dropped terms. If they really are smaller than the two you kept, the balance is self-consistent and you have found a valid asymptotic regime; if a dropped term turns out to be just as big or bigger, the balance is inconsistent and must be discarded. Trying each candidate pairing in turn (and rejecting the inconsistent ones) systematically reveals every distinguished scaling the problem admits.
Dominant balance is how you discover the thickness of a boundary layer (balance the small-parameter highest-derivative term against the dominant remaining term and solve for the layer width), the correct inner scaling in matched expansions, the leading large-x behaviour of an ODE solution, and the right exponent in a similarity solution of a PDE. The honest caveat: dominant balance is a heuristic for finding candidate behaviours, not a proof. A balance that is self-consistent is a strong lead, but it must still be confirmed against the full equation, and a problem can have several distinct consistent balances, each describing a different region or regime.
In epsilon y'' + y' + y = 0, balancing y' against y gives the slow outer solution; balancing epsilon y'' against y' forces a length scale x of order epsilon, which is exactly the boundary-layer thickness.
Each self-consistent pairing of dominant terms reveals one regime — here the outer region and the thin boundary layer.
A balance must be checked, not just assumed. After solving the chosen two-term balance you must verify the neglected terms really are smaller in that regime; a balance that makes a dropped term equally large is inconsistent and tells you to try a different pairing.