Nonhomogeneous Equations

A particular answer, plus everything the homogeneous part allows.

12 problems · about 24 minutes · A-CED.A.1, F-IF.B.6

NameDate

Figure — use these to answer the problems

A sinusoidal forcing produces a sinusoidal particular solution at the same frequency, but usually with a different size.

y = 2 sin(1x) + 0

the response amplitude2

the forcing frequency1

the time shown1

Figure 1

Straightforward practice. Get the method working first.

5 problems

y″ + 7y = 35. What is the constant particular solution?
Answer
2
What is the general solution of a nonhomogeneous equation?
- One particular solution plus the full homogeneous solution.
- Only the particular solution.
y″ + 3y = 12. What is the constant particular solution?
Answer
y″ + 2y = 10. What is the constant particular solution?
Answer
A forcing term of e^(3t). Should the guess be an exponential? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

The guess already solves the homogeneous equation. What do you multiply it by? Enter 1 for t, 2 for 2.
Answer
A forcing term of 0. Is the equation homogeneous? 1 yes, 0 no.
Answer
y″ + 6y = 18. What is the constant particular solution?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Match each forcing term to the shape of the guess.
Draw a line from each item on the left to its match on the right.
- A constant
- An exponential
- A sine
- A constant
- An exponential with the same rate
- A combination of sine and cosine
y″ + 4y = 0. What is the constant particular solution?
Answer
The Constant
y″ + 5y = 20. What is the constant particular solution?
Answer
The Pieces
How many pieces does the general solution of a nonhomogeneous equation have?
Answer