7.17.16 problem 16 (b)
Internal
problem
ID
[529]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
3.
Power
series
methods.
Section
3.6
(Applications
of
Bessel
functions).
Problems
at
page
261
Problem
number
:
16
(b)
Date
solved
:
Tuesday, January 28, 2025 at 02:35:36 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=x^{2}+y^{2} \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=1 \end{align*}
✓ Solution by Maple
Time used: 0.160 (sec). Leaf size: 139
dsolve([diff(y(x),x)=x^2+y(x)^2,y(0) = 1],y(x), singsol=all)
\[
y = \left \{\begin {array}{cc} -\frac {\left (\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right )-\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}\right ) x}{\left (\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )-\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}} & x <0 \\ 1 & x =0 \\ -\frac {\left (\operatorname {BesselY}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}+\operatorname {BesselJ}\left (-\frac {3}{4}, \frac {x^{2}}{2}\right ) \left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right )\right ) x}{\operatorname {BesselY}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2}+\left (-\Gamma \left (\frac {3}{4}\right )^{2}+\pi \right ) \operatorname {BesselJ}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )} & 0<x \end {array}\right .
\]
✓ Solution by Mathematica
Time used: 0.211 (sec). Leaf size: 114
DSolve[{D[y[x],x]==x^2+y[x]^2,{y[0]==1}},y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {\operatorname {Gamma}\left (\frac {3}{4}\right ) \left (x^2 \operatorname {BesselJ}\left (-\frac {5}{4},\frac {x^2}{2}\right )-x^2 \operatorname {BesselJ}\left (\frac {3}{4},\frac {x^2}{2}\right )+\operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )\right )-x^2 \operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (-\frac {3}{4},\frac {x^2}{2}\right )}{x \left (\operatorname {Gamma}\left (\frac {1}{4}\right ) \operatorname {BesselJ}\left (\frac {1}{4},\frac {x^2}{2}\right )-2 \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {BesselJ}\left (-\frac {1}{4},\frac {x^2}{2}\right )\right )}
\]