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Differential Equations · Second-Order Equations

Homogeneous Solutions — Practice Problems and Worked Solutions

Build and fit exponential solution families.

18 worked problems

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

1.
Choose the solution satisfying the initial data.
$y^{''} + (- 3) y^{'} + (2) y = 0, y (0) = 1, y^{'} (0) = 0$
Answer: $- e^{2 x} + 2 e^{x}$
Step-by-step solution
1. $r = 1, 2$
2. $y = C_{1} e^{1 x} + C_{2} e^{2 x}$
3. $y = - e^{2 x} + 2 e^{x}$
2.
Choose the solution satisfying the initial data.
$y^{''} + (- 2) y^{'} + (- 3) y = 0, y (0) = 3, y^{'} (0) = 5$
Answer: $2 e^{3 x} + e^{- x}$
Step-by-step solution
1. $r = - 1, 3$
2. $y = C_{1} e^{- 1 x} + C_{2} e^{3 x}$
3. $y = 2 e^{3 x} + e^{- x}$
3.
Choose the solution satisfying the initial data.
$y^{''} + (- 6) y^{'} + (8) y = 0, y (0) = 2, y^{'} (0) = 10$
Answer: $3 e^{4 x} - e^{2 x}$
Step-by-step solution
1. $r = 2, 4$
2. $y = C_{1} e^{2 x} + C_{2} e^{4 x}$
3. $y = 3 e^{4 x} - e^{2 x}$
4.
Choose the solution satisfying the initial data.
$y^{''} + (1) y^{'} + (- 2) y = 0, y (0) = 4, y^{'} (0) = - 5$
Answer: $e^{x} + 3 e^{- 2 x}$
Step-by-step solution
1. $r = - 2, 1$
2. $y = C_{1} e^{- 2 x} + C_{2} e^{1 x}$
3. $y = e^{x} + 3 e^{- 2 x}$
5.
Choose the solution satisfying the initial data.
$y^{''} + (- 6) y^{'} + (5) y = 0, y (0) = - 1, y^{'} (0) = - 9$
Answer: $- 2 e^{5 x} + e^{x}$
Step-by-step solution
1. $r = 1, 5$
2. $y = C_{1} e^{1 x} + C_{2} e^{5 x}$
3. $y = - 2 e^{5 x} + e^{x}$
6.
Choose the solution satisfying the initial data.
$y^{''} + (1) y^{'} + (- 6) y = 0, y (0) = 4, y^{'} (0) = - 2$
Answer: $2 e^{2 x} + 2 e^{- 3 x}$
Step-by-step solution
1. $r = - 3, 2$
2. $y = C_{1} e^{- 3 x} + C_{2} e^{2 x}$
3. $y = 2 e^{2 x} + 2 e^{- 3 x}$
7.
Choose the solution satisfying the initial data.
$y^{''} + (- 4) y^{'} + (3) y = 0, y (0) = 2, y^{'} (0) = 4$
Answer: $e^{3 x} + e^{x}$
Step-by-step solution
1. $r = 1, 3$
2. $y = C_{1} e^{1 x} + C_{2} e^{3 x}$
3. $y = e^{3 x} + e^{x}$
8.
Choose the solution satisfying the initial data.
$y^{''} + (- 1) y^{'} + (- 2) y = 0, y (0) = 3, y^{'} (0) = 0$
Answer: $e^{2 x} + 2 e^{- x}$
Step-by-step solution
1. $r = - 1, 2$
2. $y = C_{1} e^{- 1 x} + C_{2} e^{2 x}$
3. $y = e^{2 x} + 2 e^{- x}$
9.
Choose the solution satisfying the initial data.
$y^{''} + (1) y^{'} + (- 6) y = 0, y (0) = 3, y^{'} (0) = - 4$
Answer: $e^{2 x} + 2 e^{- 3 x}$
Step-by-step solution
1. $r = 2, - 3$
2. $y = C_{1} e^{2 x} + C_{2} e^{- 3 x}$
3. $y = e^{2 x} + 2 e^{- 3 x}$
10.
Choose the solution satisfying the initial data.
$y^{''} + (- 4) y^{'} + (3) y = 0, y (0) = 1, y^{'} (0) = - 1$
Answer: $- e^{3 x} + 2 e^{x}$
Step-by-step solution
1. $r = 3, 1$
2. $y = C_{1} e^{3 x} + C_{2} e^{1 x}$
3. $y = - e^{3 x} + 2 e^{x}$
11.
Choose the solution satisfying the initial data.
$y^{''} + (- 2) y^{'} + (- 8) y = 0, y (0) = 2, y^{'} (0) = 2$
Answer: $e^{4 x} + e^{- 2 x}$
Step-by-step solution
1. $r = - 2, 4$
2. $y = C_{1} e^{- 2 x} + C_{2} e^{4 x}$
3. $y = e^{4 x} + e^{- 2 x}$
12.
Choose the solution satisfying the initial data.
$y^{''} + (3) y^{'} + (- 4) y = 0, y (0) = 4, y^{'} (0) = - 1$
Answer: $3 e^{x} + e^{- 4 x}$
Step-by-step solution
1. $r = 1, - 4$
2. $y = C_{1} e^{1 x} + C_{2} e^{- 4 x}$
3. $y = 3 e^{x} + e^{- 4 x}$
13.
Choose the solution satisfying the initial data.
$y^{''} + (- 6) y^{'} + (8) y = 0, y (0) = 0, y^{'} (0) = 2$
Answer: $e^{4 x} - e^{2 x}$
Step-by-step solution
1. $r = 4, 2$
2. $y = C_{1} e^{4 x} + C_{2} e^{2 x}$
3. $y = e^{4 x} - e^{2 x}$
14.
Choose the solution satisfying the initial data.
$y^{''} + (- 4) y^{'} + (- 5) y = 0, y (0) = 4, y^{'} (0) = 8$
Answer: $2 e^{5 x} + 2 e^{- x}$
Step-by-step solution
1. $r = - 1, 5$
2. $y = C_{1} e^{- 1 x} + C_{2} e^{5 x}$
3. $y = 2 e^{5 x} + 2 e^{- x}$
15.
Choose the solution satisfying the initial data.
$y^{''} + (- 8) y^{'} + (12) y = 0, y (0) = 4, y^{'} (0) = 20$
Answer: $3 e^{6 x} + e^{2 x}$
Step-by-step solution
1. $r = 2, 6$
2. $y = C_{1} e^{2 x} + C_{2} e^{6 x}$
3. $y = 3 e^{6 x} + e^{2 x}$
16.
Choose the solution satisfying the initial data.
$y^{''} + (2) y^{'} + (- 3) y = 0, y (0) = 3, y^{'} (0) = - 1$
Answer: $2 e^{x} + e^{- 3 x}$
Step-by-step solution
1. $r = - 3, 1$
2. $y = C_{1} e^{- 3 x} + C_{2} e^{1 x}$
3. $y = 2 e^{x} + e^{- 3 x}$
17.
Choose the solution satisfying the initial data.
$y^{''} + (- 6) y^{'} + (5) y = 0, y (0) = 3, y^{'} (0) = 11$
Answer: $2 e^{5 x} + e^{x}$
Step-by-step solution
1. $r = 5, 1$
2. $y = C_{1} e^{5 x} + C_{2} e^{1 x}$
3. $y = 2 e^{5 x} + e^{x}$
18.
Choose the solution satisfying the initial data.
$y^{''} + (1) y^{'} + (- 2) y = 0, y (0) = 5, y^{'} (0) = - 7$
Answer: $e^{x} + 4 e^{- 2 x}$
Step-by-step solution
1. $r = 1, - 2$
2. $y = C_{1} e^{1 x} + C_{2} e^{- 2 x}$
3. $y = e^{x} + 4 e^{- 2 x}$

18 worked problems

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