DE Tutorial - Part II: Nonhomogenous Second Order Equations and Their Applications
The DE Tutorial is currently being split up into different threads to make editing these posts easier.
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:
Technique 1 : Method of Undetermined Coefficients:
- Method of Undetermined Coefficients
- The "Annihilator" Method (Similar to #1)
- Variation of Parameters
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! :D
Note that the solution to the non-homogeneous DE is a linear combination of the complimentary solution (solution to the homogeneous equation) and the particular solution (solution to the non-homogeneous equation)
Solve ; ; .
Solve the homogeneous equation first.
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).
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...