Solved: PROBLEM 56E(a) The Fresnel sine integral is

Chapter 2, Problem 56E

(choose chapter or problem)

(a) The Fresnel sine integral is defined by \(S(x)=\int_{0}^{x} \sin \left(\pi t^{2} / 2\right) \ d t\). Express the solution y(x) of the initial-value problem \(y^{\prime}-\left(\sin x^{2}\right) y=0\), y(0) = 5, in terms of S(x).

(b) Use a CAS to graph the solution curve for the IVP on \((-\infty, \infty)\).

(c) It is known that \(S(x) \rightarrow \frac{1}{2}\) as \(x \rightarrow \infty\) and \(S(x) \rightarrow-\frac{1}{2}\) as \(x \rightarrow\ -\infty\) . What does the solution y(x) approach as \(x \rightarrow \infty\) ? As \(x \rightarrow\ -\infty\)

Text Transcription:

S(x)=int_0^x sin (pi t^2 / 2) dt

y^prime - (sin x^2) y=0

(-infty, infty)

S(x) rightarrow 1/2

x rightarrow infty

S(x) rightarrow -1/2

x rightarrow -infty