Internal problem ID [11069]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-\nu ^{2}+x^{2}\right ) y-\sin \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 161
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)
\[ y \left (x \right ) = \operatorname {BesselJ}\left (\nu , x\right ) c_{2} +\operatorname {BesselY}\left (\nu , x\right ) c_{1} -\frac {x \left (\operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\nu }{2}, \frac {5}{4}-\frac {\nu }{2}, \frac {3}{4}-\frac {\nu }{2}\right ], \left [\frac {3}{2}, 1-\nu , \frac {3}{2}-\nu , \frac {3}{2}-\frac {\nu }{2}\right ], -x^{2}\right ) \operatorname {BesselJ}\left (\nu , x\right ) \Gamma \left (\nu +2\right )^{2} 2^{\nu } x^{-\nu }+\pi \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {\nu }{2}, \frac {5}{4}+\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {3}{2}, \nu +1, \frac {3}{2}+\nu , \frac {3}{2}+\frac {\nu }{2}\right ], -x^{2}\right ) 2^{-\nu } x^{\nu } \left (\cot \left (\pi \nu \right ) \operatorname {BesselJ}\left (\nu , x\right )-\operatorname {BesselY}\left (\nu , x\right )\right ) \left (\nu -1\right ) \nu \left (\nu +1\right )\right )}{2 \left (\nu -1\right ) \nu \left (\nu +1\right ) \Gamma \left (\nu +2\right )} \]
✓ Solution by Mathematica
Time used: 0.404 (sec). Leaf size: 200
DSolve[x^2*y''[x]+x*y'[x]+(x^2-\[Nu]^2)*y[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \left (-2^{\nu } \operatorname {Gamma}(\nu -1) x^{1-\nu } \operatorname {BesselJ}(\nu ,x) \, _3F_4\left (\frac {1}{2}-\frac {\nu }{2},\frac {3}{4}-\frac {\nu }{2},\frac {5}{4}-\frac {\nu }{2};\frac {3}{2},1-\nu ,\frac {3}{2}-\nu ,\frac {3}{2}-\frac {\nu }{2};-x^2\right )-\frac {2^{-\nu } \csc (\pi \nu ) \operatorname {BesselJ}(-\nu ,x) \left (\pi x^{\nu +1} \, _3F_4\left (\frac {\nu }{2}+\frac {1}{2},\frac {\nu }{2}+\frac {3}{4},\frac {\nu }{2}+\frac {5}{4};\frac {3}{2},\frac {\nu }{2}+\frac {3}{2},\nu +1,\nu +\frac {3}{2};-x^2\right )+c_2 2^{\nu +1} \operatorname {Gamma}(\nu +2)\right )}{\operatorname {Gamma}(\nu +2)}+2 (c_2 \cot (\pi \nu )+c_1) \operatorname {BesselJ}(\nu ,x)\right ) \\ \end{align*}