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Problem 3.02PP
Find the Laplace transform of the following time functions:
(a)/(f)=1 +7 f
(b) fl;f) = 4 + 7f + f2 + 6(t). where 5(t) is the unit impulse function
(c) fl;0 = e - f + 2 e - 2 t + t e - 3 t
(d) =
(e) f(f) = sinh /
(e) f{f) = sinh t
Step-by-step solution
step 1 of 12
(a)
Step 2 of 12
Consider the time function / ( 0 = l+ 7 r 
Take Laplace transform.
1 7
s+ 7
Thus, the Laplace transfomi is s + 7
Step 3 of 12
(b)
Step 4 of 12
Consider the time function / ( r ) = 4 + 7 f + / * + ^ ( r )
step 5 of 12
Take Laplace transform.
4 7 2 !, 
= - + — + — +1 
s s s
«’ +45* + 7j + 2
Thus, the Laplace transfomi is +7̂ +2
Step 6 of 12
(c)
Step 7 of 12
Consider the time function f { l ) = e " + 2 e '^ +16'” 
Take Laplace transform.
1 2 1
- + -------+ -
s + \ s + 2 (5+3)'
Thus, the Laplace transfomi is 5 + 1 5 + 2 (5 + 3)*
(d)
Step 8 of 12
Consider the time function.
«/’ +2/ + l
Step 9 of 12
aiep iu or iz
Take Laplace transform.
i [ / ( / ) ] - L ( / ' ) + i ( 2 / ) + L ( l )
2! 2 I 
= - + — + - r j s
s * + 2 i+ 2
Thus, the Laplace transfomi is 5* +25+2
(e)
Step 11 of 12
Consider the time function.
/( / )* s in h r
Step 12 of 12
Take Laplace transform.
1
* 5 * - l
Thus, the Laplace transfomi is
5^-1