Method to determine the asymptotic behavior of solutions to ODEs without fully solving the equation
In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterative, in that the result obtained by performing the method once can be used as input when the method is repeated, to obtain as many terms in the asymptotic expansion as desired.[1]
The process goes as follows:
Assume that the asymptotic behavior has the form
Make an informed guess as to which terms in the ODE might be negligible in the limit of interest.
Drop these terms and solve the resulting simpler ODE.
Check that the solution is consistent with step 2. If this is the case, then one has the controlling factor of the asymptotic behavior; otherwise, one needs try dropping different terms in step 2, instead.
Repeat the process to higher orders, relying on the above result as the leading term in the solution.
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Bender, C.M.; Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers. Springer. pp. 549–568. ISBN 0-387-98931-5.
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