Ch 7.4 - 60E
Chapter 7, Problem 60E(choose chapter or problem)
In Section 6.4 we saw that \(t y^{\prime \prime}+y^{\prime}+t y=0\) is Bessel’s equation of order v = 0. In view of (22) of that section and Table 6.4.1 a solution of the initial-value problem \(t y^{\prime \prime}+y^{\prime}+t y=0\), y(0) = 1, \(y^{\prime}(0)=0\), is \(y=J_{0}(t)\). Use this result and the procedure outlined in the instructions to Problems 17 and 18 to show that
\(\mathscr{L}\left\{J_{0}(t)\right\}=\frac{1}{\sqrt{s^{2}+1}}\).
[Hint: You might need to use Problem 46 in Exercises 7.2.]
Text Transcription:
t y^prime prime+y^prime+t y=0
y^prime 0=0
y=J_0t
mathscr L {J_0t}=1/sqrt{s^2+1}}
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