Method of undetermined coefficients

In mathematics, the method of undetermined coefficients is an approach to solving certain ordinary differential equations and recurrence relations.

The general idea is to make an "educated guess" as to the form that the solution will take, and to solve for any unknown coefficients.

Example

For example, it is well known that this differential equation:

<math>\frac {dy} {dx} = y<math>

has the following general solution:

<math>y = c_1 e^x \!<math>

One might therefore hypothesize that this differential equation:

<math>\frac {dy} {dx} = y + e^{2x} \!<math>

has a general solution of the following form:

<math>y = c_1 e^x + A e^{2x} \!<math>

where A is a coefficient whose value has to be determined. By substituting this function and its derivative into the differential equation, one can solve for A:

<math>\frac {d} {dx} \left( c_1 e^x + Ae^{2x} \right) = c_1 e^x + A e^{2x} + e^{2 x} \!<math>

<math>c_1 e^x + 2 A e^{2 x} = c_1 e^x + A e^{2 x} + e^{2 x} \!<math>

<math>2 A e^{2x} = A e^{2 x} + e^{2 x} \!<math>

<math>2 A = A + 1\,\!<math>

<math>A = 1\,\!<math>

So, the general solution to this differential equation is thus:

<math>y = c_1 e^x + 1e^{2 x} \!<math>