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Differential Equations · Second-Order Equations

Forced Equations — Practice Problems and Worked Solutions

Find particular responses to external input.

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find a constant particular solution.
$y^{''} + 1 y = 1$
Answer: $1$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $1 A = 1$
3. $A = 1$
2.
Find a constant particular solution.
$y^{''} + 4 y = 2$
Answer: $\frac{1}{2}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $4 A = 2$
3. $A = \frac{1}{2}$
3.
Find a constant particular solution.
$y^{''} + 9 y = 3$
Answer: $\frac{1}{3}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $9 A = 3$
3. $A = \frac{1}{3}$
4.
Find a constant particular solution.
$y^{''} + 2 y = 2$
Answer: $1$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $2 A = 2$
3. $A = 1$
5.
Find a constant particular solution.
$y^{''} + 3 y = - 1$
Answer: $- \frac{1}{3}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $3 A = - 1$
3. $A = - \frac{1}{3}$
6.
Find a constant particular solution.
$y^{''} + 5 y = 4$
Answer: $\frac{4}{5}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $5 A = 4$
3. $A = \frac{4}{5}$
7.
Find a constant particular solution.
$y^{''} + 1 y = 6$
Answer: $6$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $1 A = 6$
3. $A = 6$
8.
Find a constant particular solution.
$y^{''} + 4 y = 8$
Answer: $2$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $4 A = 8$
3. $A = 2$
9.
Find a constant particular solution.
$y^{''} + 2 y = 6$
Answer: $3$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $2 A = 6$
3. $A = 3$
10.
Find a constant particular solution.
$y^{''} + 3 y = 9$
Answer: $3$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $3 A = 9$
3. $A = 3$
11.
Find a constant particular solution.
$y^{''} + 5 y = 10$
Answer: $2$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $5 A = 10$
3. $A = 2$
12.
Find a constant particular solution.
$y^{''} + 6 y = 3$
Answer: $\frac{1}{2}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $6 A = 3$
3. $A = \frac{1}{2}$
13.
Find a constant particular solution.
$y^{''} + 7 y = 14$
Answer: $2$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $7 A = 14$
3. $A = 2$
14.
Find a constant particular solution.
$y^{''} + 8 y = 4$
Answer: $\frac{1}{2}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $8 A = 4$
3. $A = \frac{1}{2}$
15.
Find a constant particular solution.
$y^{''} + 9 y = 6$
Answer: $\frac{2}{3}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $9 A = 6$
3. $A = \frac{2}{3}$
16.
Find a constant particular solution.
$y^{''} + 2 y = - 4$
Answer: $- 2$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $2 A = - 4$
3. $A = - 2$
17.
Find a constant particular solution.
$y^{''} + 10 y = 5$
Answer: $\frac{1}{2}$
Step-by-step solution
1. $T r y y_{p} = A, so y_{p}^{''} = 0.$
2. $10 A = 5$
3. $A = \frac{1}{2}$

17 worked problems

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