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# 8.2E: The Inverse Laplace Transform (Exercises)

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[exer:8.2.1] 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. $${s^2-4s+3\over(s^2-4s+5)^2}$$

[exer:8.2.2] Use Theorem [thmtype: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. $${3\over(s+2)^2}-{2s+6\over s^2+4}$$

[exer:8.2.3] 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. $${3+(s-3)(2s^2+s-21)\over(s-3)(s-1)(s+4)(s-2)}$$

[exer:8.2.4] 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. $${3s+2\over(s^2+1)(s-1)^2}$$

[exer:8.2.5] Use the method of Example [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. $${2s-1\over(4s^2+1)(9s^2+1)}$$

[exer:8.2.6] 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. $${20s+40\over(4s^2-4s+5)(4s^2+4s+5)}$$

[exer:8.2.7] 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. $${2s-2\over(s-2)(s^2+2s+10)}$$

[exer:8.2.8] 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. $${5s-15\over(s^2-4s+13)(s-2)(s-1)}$$\

[exer:8.2.9] Given that $$f(t)\leftrightarrow F(s)$$, find the inverse Laplace transform of $$F(as-b)$$, where $$a>0$$.

[exer:8.2.10]

If $$s_1$$, $$s_2$$, …, $$s_n$$ are distinct and $$P$$ is a polynomial of degree less than $$n$$, then

${P(s)\over(s-s_1)(s-s_2)\cdots(s-s_n)}= {A_1\over s-s_1}+{A_2\over s-s_2}+\cdots+{A_n\over s-s_n}.$

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.

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

$F(s)={P(s)\over(s-s_1)Q_1(s)}$

is $$P(s_1)/Q_1(s_1)$$.

Explain how the results of (a) and (b) are related.