In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

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 In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...

as desired. The process goes as follows: # Assume that the asymptotic behavior has the form #:

y(x) \sim e^.

# 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.

Example

For arbitrary constants and , consider ::

xy''+(c-x)y'-ay=0.

This differential equation cannot be solved exactly. However, it is useful to consider how the solutions behave for large : it turns out that

y

behaves like

e^x\,

as ''x'' → ∞ . More rigorously, we will have

\log(y)\sim

, not

y\sim e^

. Since we are interested in the behavior of in the large limit, we change variables to = exp(''S''(''x'')), and re-express the ODE in terms of ''S''(''x''), ::

xS''+xS'^2+(c-x)S'-a=0,\,

or ::

S''+S'^2+\left(\frac-1\right)S'-\frac=0\,

where we have used the

product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...

and

chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...

to evaluate the derivatives of . Now ''suppose'' first that a solution to this ODE satisfies ::

S'^2\sim S',

as ''x'' → ∞, so that ::

S'',~\fracS',~\frac=o(S'^2),~o(S')\,

as ''x'' → ∞. Obtain then the dominant asymptotic behaviour by setting ::

S_0'^2=S_0'.

S_0

satisfies the above asymptotic conditions, then the above assumption is consistent. The terms we dropped will have been negligible with respect to the ones we kept.

S_0

is not a solution to the ODE for , but it represents ''the dominant asymptotic behavior'', which is what we are interested in. Check that this choice for

S_0

is consistent, ::

\begin
             S_0' &= 1 \\
           S_0'^2 &= 1 \\
            S_0'' &= 0 = o(S_0') \\
  \fracS_0' &= \frac = o(S_0') \\
      \frac &= o(S_0')
\end

Everything is indeed consistent. Thus the dominant asymptotic behaviour of a solution to our ODE has been found, ::

\begin
      S_0 &\sim x \\
  \log(y) &\sim x.
\end

By convention, the full asymptotic series is written as ::

y\sim Ax^p e^\left(1 + \frac + \frac + \cdots + \frac + o\left(\frac\right)\right),

so to get at least the first term of this series we have to take a further step to see if there is a power of out the front. Proceed by introducing a new subleading dependent variable, ::

S(x)\equiv S_0(x)+C(x)\,

and then seek asymptotic solutions for ''C''(''x''). Substituting into the above ODE for ''S''(''x'') we find ::

C''+C'^2+C'+\fracC'+\frac=0.

Repeating the same process as before, we keep and to find that ::

C_0=\log x^.

The leading asymptotic behaviour is then ::

y\sim x^e^x.

References

{{refs Asymptotic analysis Mathematical series Ordinary differential equations

Example

See also

References