Nonhomogeneous Equations

A particular answer, plus everything the homogeneous part allows.

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

What this lesson teaches

The student solves nonhomogeneous equations using undetermined coefficients.

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.

Straightforward practice. Get the method working first.

5 problems

y″ + 7y = 35. What is the constant particular solution?
Answer 5
Why 5.
What is the general solution of a nonhomogeneous equation?
Answer One particular solution plus the full homogeneous solution.
Why Particular plus complementary.
y″ + 3y = 12. What is the constant particular solution?
Answer 4
Why 3k = 12.
y″ + 2y = 10. What is the constant particular solution?
Answer 5
Why 2k = 10.
A forcing term of e^(3t). Should the guess be an exponential? 1 yes, 0 no.
Answer 1
Why Match the shape.

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 1
Why A factor of t.
A forcing term of 0. Is the equation homogeneous? 1 yes, 0 no.
Answer 1
Why Zero on the right.
y″ + 6y = 18. What is the constant particular solution?
Answer 3
Why 6k = 18.

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

4 problems

Match each forcing term to the shape of the guess.
Answer A constant → A constant; An exponential → An exponential with the same rate; A sine → A combination of sine and cosine
Why A sine guess needs a cosine alongside it.
y″ + 4y = 0. What is the constant particular solution?
Answer 0
Why The equation is already homogeneous.
The Constant: y″ + 5y = 20. What is the constant particular solution?
Answer 4
Why 4.
The Pieces: How many pieces does the general solution of a nonhomogeneous equation have?
Answer 2
Why 2.