Please focus on parts 42b and 42c. See pic 1 for the question and pic 2 for Proposition 2.1. Thank you. 42. Here is an interesting way to compute the Laplace trans-form of cos wt.(a) Using only Definition 1.1, show that L{sin t}(s)1/(s² + 1).(b) Suppose that f (t) has Laplace transform F(s). ShowthatLIf {at}{x) = =F (:)1L{f (at)}(s)аUse this property and the result found in part (a) toshow thatL{sin wt}(s) =s2 + w² •(c) If ƒ (t) = sin wt, then f'(t) = w cos wt. Use Proposi-tion 2.1 to show that= w coS wt. Use Proposi-SWL{wcos wt}(s) =s² + w² 'thus ensuringL{cos wt}(s)||s² + w? 5.2 Basic Properties This section will discuss the most important properties of the Laplace transform.of the Laplace We will state each property in a proposition.TransformThe Laplace transform of derivativesThe most important property for our purposes is the relationship between the Laplacetransform and the derivative. As we will see, the next proposition is the key toolwhen using the Laplace transform to solve differential equations.PROPOSITION 2.1Suppose y is a piecewise differentiable function of exponential order. Suppose alsothat y' is of exponential order. Then for large values of s,L(y')(s) = s L(y)(s) – y(0) = sY (s) – y(0),(2.2)where Y (s) is the Laplace transform of y.

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Description: Please focus on parts 42b and 42c. See pic 1 for the question and pic 2 for Proposition 2.1. Thank you. 42. Here is an interesting way to compute the Laplace trans-form of cos wt.(a) Using only Definition 1.1, show that L{sin t}(s)1/(s² + 1).(b) Supp

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42. Here is an interesting way to compute the Laplace trans- form of cos wt. (a) Using only Definition 1.1, show that L{sin t}(s) 1/(s² + 1). (b) Suppose that f (t) has Laplace transform F(s). Show that (;). L{f(at)}(s) = F - а a Use this property and the result found in part (a) to show that L{sin wt}(s) = s² + w² ° (c) If f (t) = sin wt, then f'(t) tion 2.1 to show that = w cos wt. Use Proposi- L{wcos wt}(s) = s2 + w² ' thus ensuring S L{cos wt}(s) : s2 + w² °

42. Here is an interesting way to compute the Laplace trans- form of cos wt. (a) Using only Definition 1.1, show that L{sin t}(s) 1/(s² + 1). (b) Suppose that f (t) has Laplace transform F(s). Show that (;). L{f(at)}(s) = F - а a Use this property and the result found in part (a) to show that L{sin wt}(s) = s² + w² ° (c) If f (t) = sin wt, then f'(t) tion 2.1 to show that = w cos wt. Use Proposi- L{wcos wt}(s) = s2 + w² ' thus ensuring S L{cos wt}(s) : s2 + w² °

Advanced Engineering Mathematics

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ISBN:9780470458365

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Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Please focus on parts 42b and 42c. See pic 1 for the question and pic 2 for Proposition 2.1. Thank you.

42. Here is an interesting way to compute the Laplace trans-
form of cos wt.
(a) Using only Definition 1.1, show that L{sin t}(s)
1/(s² + 1).
(b) Suppose that f (t) has Laplace transform F(s). Show
that
LIf {at}{x) = =F (:)
1
L{f (at)}(s)
а
Use this property and the result found in part (a) to
show that
L{sin wt}(s) =
s2 + w² •
(c) If ƒ (t) = sin wt, then f'(t) = w cos wt. Use Proposi-
tion 2.1 to show that
= w coS wt. Use Proposi-
SW
L{wcos wt}(s) =
s² + w² '
thus ensuring
L{cos wt}(s)
||
s² + w?

5.2 Basic Properties This section will discuss the most important properties of the Laplace transform.
of the Laplace We will state each property in a proposition.
Transform
The Laplace transform of derivatives
The most important property for our purposes is the relationship between the Laplace
transform and the derivative. As we will see, the next proposition is the key tool
when using the Laplace transform to solve differential equations.
PROPOSITION 2.1
Suppose y is a piecewise differentiable function of exponential order. Suppose also
that y' is of exponential order. Then for large values of s,
L(y')(s) = s L(y)(s) – y(0) = sY (s) – y(0),
(2.2)
where Y (s) is the Laplace transform of y.

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