A nonhomogeneous equation has something other than zero on the right: ay″ + by′ + cy = g(t). That g is the forcing term.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The structure of the answer
The general solution is one particular solution plus the whole homogeneous solution. Two separate pieces, added.
Why that works
The difference of any two particular solutions solves the homogeneous equation. So one particular solution plus the homogeneous family covers everything.
Undetermined coefficients
Guess a particular solution of the same shape as the forcing term, with unknown coefficients, then substitute and solve for them.
What shape to guess
A polynomial forcing wants a polynomial guess. An exponential wants an exponential. A sine or cosine wants a combination of both.
When the guess clashes
If the guess already solves the homogeneous equation, multiply it by t. Otherwise it contributes nothing.
Particular plus homogeneous
The general solution of a nonhomogeneous equation is any particular solution plus the general solution of the homogeneous version. The structure separates the forcing from the system's own behaviour.
Undetermined coefficients
For simple forcing terms, guess a solution of the same form with unknown coefficients and substitute to determine them. It is quick and works only for polynomial, exponential and sinusoidal forcing.
When the guess collides
If the guessed form already solves the homogeneous equation, it must be multiplied by t. That collision is the algebraic signature of resonance, which the next chapter treats physically.
Variation of parameters is general
It handles any forcing term, at the cost of integrals that may be awkward. Having both methods means choosing the quick one when it applies and the general one when it does not.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 1.68)
Step 3: Watch an Example
One step at a time.
Watch Rosa Guess a Constant
Rosa solves y″ + 4y = 12.
- Step 1
She notices the forcing term is a constant, so she guesses a constant particular solution y = k.
Step 4: Your Turn
Practice makes it stick.
The Constant
Problem 1 of 2
y″ + 5y = 20. What is the constant particular solution?
The Pieces
Problem 2 of 2
How many pieces does the general solution of a nonhomogeneous equation have?
Guess and Check
1 of 8
y″ + 3y = 12. What is the constant particular solution?
2 of 8
y″ + 2y = 10. What is the constant particular solution?
3 of 8
A forcing term of e^(3t). Should the guess be an exponential? 1 yes, 0 no.
4 of 8
The guess already solves the homogeneous equation. What do you multiply it by? Enter 1 for t, 2 for 2.
5 of 8
A forcing term of 0. Is the equation homogeneous? 1 yes, 0 no.
6 of 8
y″ + 6y = 18. What is the constant particular solution?
7 of 8
Match each forcing term to the shape of the guess.
Tap a card on the left to start.
8 of 8
y″ + 4y = 0. What is the constant particular solution?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y″ + 7y = 35. What is the constant particular solution?
Question 2 of 2
What is the general solution of a nonhomogeneous equation?
What You Learned
- A nonhomogeneous equation has a forcing term on the right-hand side.
- Its general solution is one particular solution plus the whole homogeneous family.
- Guess a particular solution shaped like the forcing term, and multiply by t if it clashes.