Q:

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.

Accepted Solution

A:
Answer:a. [tex]\frac {2} {s-1}[/tex] converges to s> 1.b. [tex]\frac{3}{e^3 \left(s-5 \right)}[/tex] converges to s> 5.c. [tex]- \frac {2}{s + 3}[/tex] converges to s> - 3.d. [tex]\frac {s}{s^2 + 25}[/tex] converges to s> 0.e. [tex]\frac {10} {s^2 + 1}[/tex] converges even s> 0.f. [tex]\frac {12}{s^2 + 4}[/tex] converges to s> 0.g. [tex]-\frac {5\left(\cos\left (1\right) s-2 \sin\left(1\right)\right)}{s^2 + 4}[/tex] converges to s> 0.h. [tex]\frac {1} {s ^ 2 + 4}[/tex] converges to s> 0.Step-by-step explanation:a. [tex]L \left\{2e^t \right\} = 2L \left\{e^t \right\} = 2 \cdot \frac {1} {s-1} = \frac {2} {s-1}[/tex] converges to s> 1.b. [tex]L \left\{3e^{5t-3} \right\} = 3e^{-3} L \left\{e^{5t} \right\} = 3e^{-3} L \left\{e^{5t} \right\} = \frac{3}{e^3 \left(s-5 \right)}[/tex] converges to s> 5.c. [tex]L \left\{-2e^{-3t} \right\} = -2L \left\{e^{-3t} \right\} = - \frac {2}{s + 3}[/tex] converges to s> - 3.d. [tex]L \left\{\cos\left (5t \right)\right\} = \frac {s}{s^2 + 25}[/tex] converges to s> 0.e. [tex]L \left\{10 \sin\left(t\right)\right\} = 10L\left\{\sin\left(t\right)\right\} = \frac {10} {s^2 + 1}[/tex] converges even s> 0.f. [tex]L \left\{6\sin \left(2t \right) \right\} = 6L\left\{\sin\left (2t\right)\right\} = \frac {12}{s^2 + 4}[/tex] converges to s> 0.g. [tex]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}[/tex] converges to s> 0.h. [tex]L\left\{\sin \left(t\right)\cos \left(t\right)\right\} = L\left\{\sin\left(2t\right)\frac{1}{2}\right\} =\frac{1}{2}\cdot \frac{2}{s^2+4} = \frac {1} {s ^ 2 + 4}[/tex] converges to s> 0.