Question

Find the Laplace transformation of each of the following functions. In each case, specify the values of s for which the integral converges. 2et a. b. 3e 5t-3 C. 2/-e 3 cos 5 10sin 6t 6sin 2t -5 cos 2 (P+1) (sin-cost) d. e. f. g. h. i.

Answer 1

a.
`\frac {2} {s-1}` converges to s> 1.

b.
`(3)/(e^3 \left(s-5 \right))` converges to s> 5.

c.
`- \frac {2}{s + 3}` converges to s> - 3.

d.
`\frac {s}{s^2 + 25}` converges to s> 0.

e.
`\frac {10} {s^2 + 1}` converges even s> 0.

f.
`\frac {12}{s^2 + 4}` converges to s> 0.

g.
`-\frac {5\left(\cos\left (1\right) s-2 \sin\left(1\right)\right)}{s^2 + 4}` converges to s> 0.

h.
`\frac {1} {s ^ 2 + 4}` converges to s> 0.

Explanation:

a.
`L \left\{2e^t \right\} = 2L \left\{e^t \right\} = 2 \cdot \frac {1} {s-1} = \frac {2} {s-1}` converges to s> 1.

b.
`L \left\{3e^(5t-3) \right\} = 3e^(-3) L \left\{e^(5t) \right\} = 3e^(-3) L \left\{e^(5t) \right\} = (3)/(e^3 \left(s-5 \right))` converges to s> 5.

c.
`L \left\{-2e^(-3t) \right\} = -2L \left\{e^(-3t) \right\} = - \frac {2}{s + 3}` converges to s> - 3.

d.
`L \left\{\cos\left (5t \right)\right\} = \frac {s}{s^2 + 25}` converges to s> 0.

e.
`L \left\{10 \sin\left(t\right)\right\} = 10L\left\{\sin\left(t\right)\right\} = \frac {10} {s^2 + 1}` converges even s> 0.

f.
`L \left\{6\sin \left(2t \right) \right\} = 6L\left\{\sin\left (2t\right)\right\} = \frac {12}{s^2 + 4}` converges to s> 0.

g.
`L \left\{-5\cos\left(2t + 1\right) \right\} = -5L\left\{\cos\left(2t + 1 \right)\right\} = -\frac {5\left(\cos\left (1\right) s-2 \sin\left(1\right)\right)}{s^2 + 4}` converges to s> 0.

h.
`L\left\{\sin \left(t\right)\cos \left(t\right)\right\} = L\left\{\sin\left(2t\right)(1)/(2)\right\} =(1)/(2)\cdot (2)/(s^2+4) = \frac {1} {s ^ 2 + 4}` converges to s> 0.