Cos L S2
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
Part A) Let f(x) = sin (√(x)), use the fact that f'(x) = (cos(√(x)))/(2(√(x))) and f(0) = 0 to show what is depicted in the picture below that contains the laplance cos(√(x))/(√(x)).
Part B) By taking the laplace transform of the above series, show what is depicted in the picture below containing the laplace sin(√(x)) .
![The image contains a mathematical expression involving a Laplace transform:
\[
\mathcal{L} \left( \frac{\cos \sqrt{x}}{\sqrt{x}} \right) = \frac{\sqrt{\pi}}{s^{\frac{1}{2}}} e^{-\frac{1}{4s}}
\]
This equation shows the Laplace transform of the function \(\frac{\cos \sqrt{x}}{\sqrt{x}}\). On the right side, the result is expressed in terms of \(s\), incorporating the square root of \(\pi\), a half power of \(s\), and the exponential function \(e\) raised to a negative fractional exponent \(-\frac{1}{4s}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40884399-19a4-4ad9-a712-30c31bac2ede%2F978ab9b7-31a4-49a5-be3d-235c6dd523a1%2Fdm2xgv6r_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression involving a Laplace transform:
\[
\mathcal{L} \left( \frac{\cos \sqrt{x}}{\sqrt{x}} \right) = \frac{\sqrt{\pi}}{s^{\frac{1}{2}}} e^{-\frac{1}{4s}}
\]
This equation shows the Laplace transform of the function \(\frac{\cos \sqrt{x}}{\sqrt{x}}\). On the right side, the result is expressed in terms of \(s\), incorporating the square root of \(\pi\), a half power of \(s\), and the exponential function \(e\) raised to a negative fractional exponent \(-\frac{1}{4s}\).
![The image presents a mathematical expression related to the Laplace transform:
\[
\mathcal{L}(\sin \sqrt{t}) = \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} e^{\frac{-1}{4s}}
\]
This formula represents the Laplace transform of the function \(\sin \sqrt{t}\), where \(\mathcal{L}\) denotes the Laplace transformation operator. The result is expressed in terms of \(s\), a complex frequency variable often used in the Laplace domain. The expression involves an exponential term and fractional powers of \(s\) and \(\pi\), emphasizing the importance of these components in the transformation process.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F40884399-19a4-4ad9-a712-30c31bac2ede%2F978ab9b7-31a4-49a5-be3d-235c6dd523a1%2F50mhsb_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a mathematical expression related to the Laplace transform:
\[
\mathcal{L}(\sin \sqrt{t}) = \frac{\sqrt{\pi}}{2s^{\frac{3}{2}}} e^{\frac{-1}{4s}}
\]
This formula represents the Laplace transform of the function \(\sin \sqrt{t}\), where \(\mathcal{L}\) denotes the Laplace transformation operator. The result is expressed in terms of \(s\), a complex frequency variable often used in the Laplace domain. The expression involves an exponential term and fractional powers of \(s\) and \(\pi\), emphasizing the importance of these components in the transformation process.
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