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Second Order NON-Homogeneous Differential Equations

Ah...the dreadful second order non-homogeneous differential equation has the form:

I will go through three techniques on how to solve these nasties:

- Method of Undetermined Coefficients
- The "Annihilator" Method (Similar to #1)
- Variation of Parameters
Technique 1 : Method of Undetermined Coefficients:

In the case we have a differential equation like , we can guess what the particular solution to a DE may be, depending on what is. For example, let us say that . We would assume that a particular solution to the DE would be . To find the Undetermined Coefficients, plug back into the original DE.

If , we assume that the particular solution to the DE would have the form of . We too would substitute into the DE to find the unknown coefficient value.

If , we assume that the particular solution to the DE would have the form of . Again, to find the unknown coefficients, substitute into the original DE.

Sometimes, the guess of isn't that obvious. Try to think outside the box when solving these problems!

Note that the solution to the non-homogeneous DE is a linear combination of thecomplimentary solution(solution to the homogeneous equation) and theparticular solution(solution to the non-homogeneous equation)

Example 16:

Solve ; ; .

Solve the homogeneous equation first.

.

Thus, .

Now solve the non-homogeneous equation.

.

Using the method of Undetermined Coefficients, we guess and assume that the particular solution will have the form:

.

Substituting these values into the original DE, we get:

Now compare the coefficients (like in Partial Fractions)

Solving for the constants, we get , and (verify).

Thus, .

Therefore, the solution is

.

Find so we can apply the initial conditions:

.

Apply the initial conditions.

.

Solving for , get (verify)

Therefore, the solution to the initial value problem is:

.

I will discuss the next two sometime later...