Learning Materials
Features
Discover
Chapter 13: Q25E (page 561)
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)
The obtained integral is \(\int_0^x r {J_0}(r)dr = x{J_1}(x)\).
Over 30 million students worldwide already upgrade their learning with 91影视!
All the tools & learning materials you need for study success - in one app.
Get started for free
Use the recurrence relation in Problem 28 along with (26) and (27) to express \({J_{3/2}}(x),{J_{5/2}}(x),{J_{ - 3/2}}(x),{J_{ - 5/2}}(x)\) in terms of \(sinx,cosx\), and powers of .
Show that \(y = {x^{1/2}}w\left( {\frac{2}{3}\alpha {x^{3/2}}} \right)\) is a solution of the given form of Airy鈥檚 differential equation whenever w is a solution of the indicated Bessel鈥檚 equation. (Hint: After differentiating, substituting, and simplifying, then let \(t = \frac{2}{3}\alpha {x^{3/2}}\))
(a) \(y'' + {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' + \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)
(b) \(y'' - {\alpha ^2}xy = 0,x > 0;{t^2}w'' + tw' - \left( {{t^2} - \frac{1}{9}w} \right) = 0,t > 0\)
Use the result in parts (a) and (b) of Problem 36 to express the general solution on \((0,\infty )\) of each of the two forms of Airy鈥檚 equation in terms of Bessel functions.
(a) Use the general solution given in Example 5 to solve the IVP \(4x'' + {e^{ - 0.1t}}x = 0,x(0) = 1,x'(0) = - \frac{1}{2}\). Also use \(J_0^'(x) = - {J_1}(x)\) and \(Y_0^'(x) = - {Y_1}(x)\) along with Table 6.4.1 or a CAS to evaluate coefficients.
(b) Use a CAS to graph the solution obtained in part (a) for \(0 \le t < \infty \).
In Problems 29 and 30 use (22) or (23) to obtain the given result.
\(\int_0^x r {J_0}(r)dr = x \times {J_1}(x)\)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine