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Differential Equations · Transforms & Systems

Linear ODE Systems — Practice Problems and Worked Solutions

Use matrix exponentials to evolve coupled states.

18 worked problems

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

1.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [1002], X (0) = [11]$
Answer: $[e^{x} e^{2 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{x} e^{2 x}]$
2.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [- 1 0 03], X (0) = [2 - 1]$
Answer: $[2 e^{- x} - e^{3 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [2 e^{- x} - e^{3 x}]$
3.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [20 0 - 2], X (0) = [13]$
Answer: $[e^{2 x} 3 e^{- 2 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{2 x} 3 e^{- 2 x}]$
4.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [0 - 1 10], X (0) = [10]$
Answer: $[cos (x) - sin (x)]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [cos (x) - sin (x)]$
5.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [01 - 1 0], X (0) = [02]$
Answer: $[- 2 sin (x) 2 cos (x)]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [- 2 sin (x) 2 cos (x)]$
6.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [3001], X (0) = [- 1 2]$
Answer: $[- e^{3 x} 2 e^{x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [- e^{3 x} 2 e^{x}]$
7.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [2001], X (0) = [12]$
Answer: $[e^{2 x} 2 e^{x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{2 x} 2 e^{x}]$
8.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [- 2 0 04], X (0) = [31]$
Answer: $[3 e^{- 2 x} e^{4 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [3 e^{- 2 x} e^{4 x}]$
9.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [10 0 - 3], X (0) = [22]$
Answer: $[2 e^{x} 2 e^{- 3 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [2 e^{x} 2 e^{- 3 x}]$
10.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [0 - 2 20], X (0) = [11]$
Answer: $[sin (2 x) + cos (2 x) - sin (2 x) + cos (2 x)]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [sin (2 x) + cos (2 x) - sin (2 x) + cos (2 x)]$
11.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [4002], X (0) = [1 - 1]$
Answer: $[e^{4 x} - e^{2 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{4 x} - e^{2 x}]$
12.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [- 1 0 0 - 2], X (0) = [23]$
Answer: $[2 e^{- x} 3 e^{- 2 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [2 e^{- x} 3 e^{- 2 x}]$
13.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [5003], X (0) = [11]$
Answer: $[e^{5 x} e^{3 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{5 x} e^{3 x}]$
14.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [0 - 3 30], X (0) = [20]$
Answer: $[2 cos (3 x) - 2 sin (3 x)]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [2 cos (3 x) - 2 sin (3 x)]$
15.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [1005], X (0) = [3 - 2]$
Answer: $[3 e^{x} - 2 e^{5 x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [3 e^{x} - 2 e^{5 x}]$
16.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [- 4 0 01], X (0) = [14]$
Answer: $[e^{- 4 x} 4 e^{x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [e^{- 4 x} 4 e^{x}]$
17.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [30 0 - 1], X (0) = [21]$
Answer: $[2 e^{3 x} e^{- x}]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [2 e^{3 x} e^{- x}]$
18.
Solve the linear system with the given initial vector.
$X^{'} = A X, A = [0 - 4 10], X (0) = [12]$
Answer: $[sin (2 x) + cos (2 x) - 2 sin (2 x) + 2 cos (2 x)]$
Step-by-step solution
1. $X (x) = e^{A x} X (0)$
2. $X (x) = [sin (2 x) + cos (2 x) - 2 sin (2 x) + 2 cos (2 x)]$

18 worked problems

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