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Chapter 12: Q7P (page 597)

Determine the solution of the differential equations.
(a) $x y^{'}' = y$
(b) $y^{'}' - x^{4} y = 0$

Short Answer

Expert verified

(a) The solution is $y = x^{\frac{1}{2}} Z_{1} (2 i x^{\frac{1}{2}})$ .

(b) The solution is $y = x^{\frac{1}{2}} Z_{\frac{1}{6}} (\frac{1}{3} 3 i x^{3}) .$ .

Step by step solution

01

Concept of Bessel functions by using equations (16.1) and (16.2):

Consider the standard form is,

$y^{'}' + \frac{1 - 2 a}{x y^{'}} + (b c x^{c - 1})^{2} + [\frac{a^{2} - p^{2} c^{2}}{x^{2}}] y = 0$

Solution to the above mentioned standard equation is,

$y = x^{a} Z_{p} (b x^{c})$

02

(a) Find the solutions of differential equation:

Consider the differential equation as below.

$x y^{'}' = y'' y^{'}'' - \frac{y}{X} = 0$

Compare the equation with standard Bessel function equation as follows:

$a = \frac{1}{2} c = \frac{1}{2} p = 1 b = 2 i$

So,

role="math" localid="1659268440330" $y = x^{\frac{1}{2}} Z_{1} (2 i x^{\frac{1}{2}})$

Now,

$y = x^{\frac{1}{2}} Z_{1} (2 i x^{\frac{1}{2}}) y = x^{\frac{1}{2}} I_{1} (2 i x^{\frac{1}{2}}) (鈭� I_{1} = i^{(} - 1) I_{1} (i x))$

03

(b) Find the solutions of y''-x4y=0 differential equation:

Consider the differential equation as below.

$y^{'}' - x^{4} y = 0$

Then, $y^{'}' - \frac{y}{X} = 0$ .

Compare the equation with standard Bessel function equation as follows:

$a = \frac{1}{2} c = 3 p = \frac{1}{6} b = \frac{1}{3} i$

So,

$y = x^{\frac{1}{2}} Z_{\frac{1}{6}} (\frac{1}{3} 3 i x^{3}) .$

Now,

$y = x^{\frac{1}{2}} Z_{\frac{1}{6}} (\frac{1}{3} 3 i x^{3}) y = x^{\frac{1}{2}} I_{\frac{1}{6}} (\frac{1}{3} x^{3}) (鈭� I_{1} = i^{- 1} I_{1} (i x))$

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Most popular questions from this chapter

To calculate the given system of equation.

$\frac{d}{dx} [{(x)}^{p} J_{p} (x)] = {(x)}^{p} J_{p - 1} (x)$

Find the best (in the least squares sense) second-degree polynomial approximation to each of the given functions over the interval -1<x<1.

x⁴

To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval $j2(x)$

Verify equations (10.3) and (10.4).

(10.3) : (1-x²) u"-2 (m+1) xu'+[l(l+1) - m(m+1)] u=0

(10.4) : (1-x²) (u')" -2 [(m+1)+1] x(u')'+ [l(l+1) - (m+1)(m+2)]u'=0

To show $J_{3 / 2} (x) = x^{- 1} J_{1 / 2} (x)$ .

See all solutions

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