68.2.8 problem Problem 3.12
Internal
problem
ID
[14153]
Book
:
Differential
Equations,
Linear,
Nonlinear,
Ordinary,
Partial.
A.C.
King,
J.Billingham,
S.R.Otto.
Cambridge
Univ.
Press
2003
Section
:
Chapter
3
Bessel
functions.
Problems
page
89
Problem
number
:
Problem
3.12
Date
solved
:
Tuesday, January 28, 2025 at 06:16:07 AM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (-\nu ^{2}+x^{2}\right ) y&=\sin \left (x \right ) \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 158
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-nu^2)*y(x)=sin(x),y(x), singsol=all)
\[
y = -\frac {x^{-\nu +1} 2^{\nu -1} \operatorname {BesselJ}\left (\nu , x\right ) \Gamma \left (\nu +2\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\nu }{2}, \frac {3}{4}-\frac {\nu }{2}, \frac {5}{4}-\frac {\nu }{2}\right ], \left [\frac {3}{2}, -\nu +1, \frac {3}{2}-\nu , \frac {3}{2}-\frac {\nu }{2}\right ], -x^{2}\right )}{\nu \left (\nu -1\right ) \left (\nu +1\right )}+\operatorname {BesselJ}\left (\nu , x\right ) c_{2} +\operatorname {BesselY}\left (\nu , x\right ) c_{1} -\frac {\pi 2^{-\nu -1} x^{\nu +1} \left (\cot \left (\pi \nu \right ) \operatorname {BesselJ}\left (\nu , x\right )-\operatorname {BesselY}\left (\nu , x\right )\right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}+\frac {\nu }{2}, \frac {3}{4}+\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {3}{2}, \nu +1, \frac {3}{2}+\nu , \frac {3}{2}+\frac {\nu }{2}\right ], -x^{2}\right )}{\Gamma \left (\nu +2\right )}
\]
✓ Solution by Mathematica
Time used: 0.229 (sec). Leaf size: 72
DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-\[Nu]^2)*y[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \operatorname {BesselJ}(\nu ,x) \int _1^x-\frac {\pi \operatorname {BesselY}(\nu ,K[1]) \sin (K[1])}{2 K[1]}dK[1]+\operatorname {BesselY}(\nu ,x) \int _1^x\frac {\pi \operatorname {BesselJ}(\nu ,K[2]) \sin (K[2])}{2 K[2]}dK[2]+c_1 \operatorname {BesselJ}(\nu ,x)+c_2 \operatorname {BesselY}(\nu ,x)
\]