Method of Undetermined Coefficients

where are constants, , and is a given function called the nonhomogenous term The general solution structure is as follows:

where are solutions of the homogeneous equation:

where is a particular solution of the nonhomogenous (original) equation:

Reminder: is the Characteristic Equation for the family of solutions

Approach

Step 1: Find complementary solutions Step 2: Find particular solution Constructing a trial term:

where is the number of terms in that have , where is a n-degree polynomial If any term in the guess for is a solution to , multiply by . In other words, multiply by If there are multiple terms, superposition applies. What that means is you substitute one trial solution terms back into the original equation, at a time, in order to find the undetermined coefficients. Note that trig functions are actually exponentials:

Also recall

Euler's Identity

Link to original

Plug that term into the original equation to solve for undetermined coefficients. Step 3:

Example

Step 1: Find complementary solutions Step 2: Find particular solution Plug in: Step 3:

Now do the IVP. We have already concluded the method of undetermined coefficients. After doing the IVP you get