68.2.8 problem Problem 3.12

Internal problem ID [14082]
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 : Monday, March 31, 2025 at 12:02:58 PM
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*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 158

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(-nu^2+x^2)*y(x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x^{1-\nu } 2^{\nu -1} \operatorname {BesselJ}\left (\nu , x\right ) \Gamma \left (\nu +2\right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}-\frac {\nu }{2}, \frac {3}{4}-\frac {\nu }{2}, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}, 1-\nu , \frac {3}{2}-\nu , \frac {3}{2}-\frac {\nu }{2}\right ], -x^{2}\right )}{\nu \left (\nu -1\right ) \left (\nu +1\right )}+\operatorname {BesselY}\left (\nu , x\right ) c_1 +\operatorname {BesselJ}\left (\nu , x\right ) c_2 -\frac {\pi 2^{-1-\nu } x^{\nu +1} \left (\cot \left (\pi \nu \right ) \operatorname {BesselJ}\left (\nu , x\right )-\operatorname {BesselY}\left (\nu , x\right )\right ) \operatorname {hypergeom}\left (\left [\frac {3}{4}+\frac {\nu }{2}, \frac {5}{4}+\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {3}{2}, \nu +1, \frac {3}{2}+\nu , \frac {3}{2}+\frac {\nu }{2}\right ], -x^{2}\right )}{\Gamma \left (\nu +2\right )} \]

✓ Mathematica. Time used: 0.229 (sec). Leaf size: 72

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-\[Nu]^2)*y[x]==Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \operatorname {BesselJ}(\nu ,x) \int _1^x-\frac {\pi \operatorname {BesselY}(\nu ,K[1]) \sin (K[1])}{2 K[1]}dK[1]+\operatorname {BesselY}(\nu ,x) \int _1^x\frac {\pi \operatorname {BesselJ}(\nu ,K[2]) \sin (K[2])}{2 K[2]}dK[2]+c_1 \operatorname {BesselJ}(\nu ,x)+c_2 \operatorname {BesselY}(\nu ,x) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (-nu**2 + x**2)*y(x) - sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (nu**2*y(x) - x**2*(y(x) + Derivative(y(x), (x, 2))) + sin(x))/x cannot be solved by the factorable group method