8.2.1: The Inverse Laplace Transform (Exercises)

Q8.2.1

1. Use the table of Laplace transforms to find the inverse Laplace transform.

1. \( {3\over(s-7)^4}\)
2. \( {2s-4\over s^2-4s+13}\)
3. \( {1\over s^2+4s+20}\)
4. \( {2\over s^2+9}\)
5. \( {s^2-1\over(s^2+1)^2}\)
6. \( {1\over(s-2)^2-4}\)
7. \( {12s-24\over(s^2-4s+85)^2}\)
8. \( {2\over(s-3)^2-9}\)
9. \( {s^2-4s+3\over(s^2-4s+5)^2}\)

2. Use Theorem 8.2.1 and the table of Laplace transforms to find the inverse Laplace transform.

1. \( {2s+3\over(s-7)^4}\)
2. \( {s^2-1\over(s-2)^6}\)
3. \( {s+5\over s^2+6s+18}\)
4. \( {2s+1\over s^2+9}\)
5. \( {s\over s^2+2s+1}\)
6. \( {s+1\over s^2-9}\)
7. \( {s^3+2s^2-s-3\over(s+1)^4}\)
8. \( {2s+3\over(s-1)^2+4}\)
9. \( {1\over s}-{s\over s^2+1}\)
10. \( {3s+4\over s^2-1}\)
11. \( {3\over s-1}+{4s+1\over s^2+9}\)
12. \( {3\over(s+2)^2}-{2s+6\over s^2+4}\)

3. Use Heaviside’s method to find the inverse Laplace transform.

1. \( {3-(s+1)(s-2)\over(s+1)(s+2)(s-2)}\)
2. \( {7+(s+4)(18-3s)\over(s-3)(s-1)(s+4)}\)
3. \( {2+(s-2)(3-2s)\over(s-2)(s+2)(s-3)}\)
4. \( {3-(s-1)(s+1)\over(s+4)(s-2)(s-1)}\)
5. \( {3+(s-2)(10-2s-s^2)\over(s-2)(s+2)(s-1)(s+3)}\)
6. \( {3+(s-3)(2s^2+s-21)\over(s-3)(s-1)(s+4)(s-2)}\)

4. Find the inverse Laplace transform.

1. \( {2+3s\over(s^2+1)(s+2)(s+1)}\)
2. \( {3s^2+2s+1\over(s^2+1)(s^2+2s+2)}\)
3. \( {3s+2\over(s-2)(s^2+2s+5)}\)
4. \( {3s^2+2s+1\over(s-1)^2(s+2)(s+3)}\)
5. \( {2s^2+s+3\over(s-1)^2(s+2)^2}\)
6. \( {3s+2\over(s^2+1)(s-1)^2}\)

5. Use the method of Example 8.2.9 to find the inverse Laplace transform.

1. \( {3s+2\over(s^2+4)(s^2+9)}\)
2. \( {-4s+1\over(s^2+1)(s^2+16)}\)
3. \( {5s+3\over(s^2+1)(s^2+4)}\)
4. \( {-s+1\over(4s^2+1)(s^2+1)}\)
5. \( {17s-34\over(s^2+16)(16s^2+1)}\)
6. \( {2s-1\over(4s^2+1)(9s^2+1)}\)

6. Find the inverse Laplace transform.

1. \( {17 s-15\over(s^2-2s+5)(s^2+2s+10)}\)
2. \( {8s+56\over(s^2-6s+13)(s^2+2s+5)}\)
3. \( {s+9\over(s^2+4s+5)(s^2-4s+13)}\)
4. \( {3s-2\over(s^2-4s+5)(s^2-6s+13)}\)
5. \( {3s-1\over(s^2-2s+2)(s^2+2s+5)}\)
6. \( {20s+40\over(4s^2-4s+5)(4s^2+4s+5)}\)

7. Find the inverse Laplace transform.

1. \( {1\over s(s^2+1)}\)
2. \( {1\over(s-1)(s^2-2s+17)}\)
3. \( {3s+2\over(s-2)(s^2+2s+10)}\)
4. \( {34-17s\over(2s-1)(s^2-2s+5)}\)
5. \( {s+2\over(s-3)(s^2+2s+5)}\)
6. \( {2s-2\over(s-2)(s^2+2s+10)}\)

8. Find the inverse Laplace transform.

1. \( {2s+1\over(s^2+1)(s-1)(s-3)}\)
2. \( {s+2\over(s^2+2s+2)(s^2-1)}\)
3. \( {2s-1\over(s^2-2s+2)(s+1)(s-2)}\)
4. \( {s-6\over(s^2-1)(s^2+4)}\)
5. \( {2s-3\over s(s-2)(s^2-2s+5)}\)
6. \( {5s-15\over(s^2-4s+13)(s-2)(s-1)}\)

9. Given that \(f(t)\leftrightarrow F(s)\), find the inverse Laplace transform of \(F(as-b)\), where \(a>0\).

10.

1. If \(s_1\), \(s_2\), …, \(s_n\) are distinct and \(P\) is a polynomial of degree less than \(n\), then \[{P(s)\over(s-s_1)(s-s_2)\cdots(s-s_n)}= {A_1\over s-s_1}+{A_2\over s-s_2}+\cdots+{A_n\over s-s_n}.\nonumber \] Multiply through by \(s-s_i\) to show that \(A_i\) can be obtained by ignoring the factor \(s-s_i\) on the left and setting \(s=s_i\) elsewhere.
2. Suppose \(P\) and \(Q_1\) are polynomials such that \(\mbox{degree}(P)\le\mbox{degree}(Q_1)\) and \(Q_1(s_1)\ne0\). Show that the coefficient of \(1/(s-s_1)\) in the partial fraction expansion of \[F(s)={P(s)\over(s-s_1)Q_1(s)}\nonumber \] is \(P(s_1)/Q_1(s_1)\).
3. Explain how the results of (a) and (b) are related.