8.2E: The Inverse Laplace Transform (Exercises)

Q8.2.1

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

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

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

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

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

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

4. Find the inverse Laplace transform.

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

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

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

6. Find the inverse Laplace transform.

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

7. Find the inverse Laplace transform.

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

8. Find the inverse Laplace transform.

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

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 $${P(s)}{(s-s_1)(s-s_2)\cdots(s-s_n)}= {A_1}{ s-s_1}+{A_2}{ s-s_2}+\cdots+{A_n}{ s-s_n}.\nonumber$$ 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 $$F(s)={P(s)}{(s-s_1)Q_1(s)}\nonumber$$ is $P(s_1)/Q_1(s_1)$.
    3. Explain how the results of (a) and (b) are related.