3.9 The Method of Undetermined Coefficients

Method of Undetermined Coefficients

A method utilized for solving a specific class of non-homogenous differential equations whose forcing functions belong to a specific class of functions.

We look at the forcing function and then select a guess function, Y_p(t), based on the forcing function of the non-homogenous equation.

Total Solution

Recall that for non-homogenous differential equations, the final solution is always the linear combination of the complementary solution and the guess function — Y_p(t).

Tables of Forcing Functions

Linear Independence and Guess Functions

IF the complementary solution contains a term that is already present in the forcing function, then our guess function must be multiplied by the lowest factor of t that enables a difference to be had.

For example,

y” -3y’ + 2y = e^t

y_c(t) = c₁e^t + c₂e²t

However, we already have e^t in our y_c AND in our forcing function, so we need to change our original guess function.

Originally, our guess function (based on the table of forcing functions) is ae^t; however, because we already have an overlap with a term in the y_c(t) and in the forcing function, we need to multiply our guess function by a factor of t in order to ensure linear independence. So, our guess function is now ate^t.