Please do 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please do 2
![(3.5)
of the
Propert
w from th
3.1 Definition and Basic Properties
2. Use the definition of the Laplace transform to find the transform of r(t) =
e-3t H(t-2).
3. Find the Laplace transform of r(t)
definitions
-
sin kt and x(t)
1
sin kt = (eikt - e-ikt), cos kt =
2i
2
Check your answers in the table.
4. Find the Laplace transform of the hyperbolic functions r(t) = sinh kt and
x(t) = cosh kt using the definitions
1
sinh kt = (ekt - e-kt), cosh kt =
1
2
a) 6+5e-2t + te³t.
b) tH(t-3).
c) cos 5t.
(eikt
9. Find the inverse transform of the
7. Use the preceding exercise to compute C[t² H (t-1)].
8. Find the Laplace transform of the following functions.
cos kt using the
(ekt + e-kt).
Check your answers in the table.
5. Derive the operational formulas (3.4) and (3.5) directly from the definition.
Hint: Change variables in the integrals.
6. Use the definition of Laplace transform to show that
L[f(t)H(ta)] = e as L[f(t + a)].
145
+ e-ikt).
foll uring functions
d) sin(2t + 7).
3e-t cosht.
f) H(tr) cos(t - π).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2Fd9bfae57-fa85-4803-8a36-3da79d031640%2Fgc2f3id_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(3.5)
of the
Propert
w from th
3.1 Definition and Basic Properties
2. Use the definition of the Laplace transform to find the transform of r(t) =
e-3t H(t-2).
3. Find the Laplace transform of r(t)
definitions
-
sin kt and x(t)
1
sin kt = (eikt - e-ikt), cos kt =
2i
2
Check your answers in the table.
4. Find the Laplace transform of the hyperbolic functions r(t) = sinh kt and
x(t) = cosh kt using the definitions
1
sinh kt = (ekt - e-kt), cosh kt =
1
2
a) 6+5e-2t + te³t.
b) tH(t-3).
c) cos 5t.
(eikt
9. Find the inverse transform of the
7. Use the preceding exercise to compute C[t² H (t-1)].
8. Find the Laplace transform of the following functions.
cos kt using the
(ekt + e-kt).
Check your answers in the table.
5. Derive the operational formulas (3.4) and (3.5) directly from the definition.
Hint: Change variables in the integrals.
6. Use the definition of Laplace transform to show that
L[f(t)H(ta)] = e as L[f(t + a)].
145
+ e-ikt).
foll uring functions
d) sin(2t + 7).
3e-t cosht.
f) H(tr) cos(t - π).
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