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

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,

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

(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,

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

(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,

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

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

Short Answer

Step by step solution

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

(a) Find the solutions of differential equation:

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

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Determine the solution of the differential equations.(a)xy''=y(b) y''-x4y=0

Short Answer

Step by step solution

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

(a) Find the solutions of differential equation:

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Fundamentals of Physics

Oscillations

Thermodynamics

Work Energy and Power

Kinematics Physics

Study anywhere. Anytime. Across all devices.

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