85.65.6 problem 7

Internal problem ID [22886]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. B Exercises at page 213
Problem number : 7
Date solved : Friday, October 03, 2025 at 08:03:34 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (r^{2}+r \right ) R^{\prime \prime }+r R^{\prime }-n \left (n +1\right ) R&=0 \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 114

ode:=(r^2+r)*diff(diff(R(r),r),r)+r*diff(R(r),r)-n*(n+1)*R(r) = 0; 
dsolve(ode,R(r), singsol=all);
 

\[ R = c_1 \,r^{\sqrt {n}\, \sqrt {n +1}} \operatorname {hypergeom}\left (\left [-\sqrt {n}\, \sqrt {n +1}, -\sqrt {n}\, \sqrt {n +1}+1\right ], \left [1-2 \sqrt {n}\, \sqrt {n +1}\right ], -\frac {1}{r}\right )+c_2 \,r^{-\sqrt {n}\, \sqrt {n +1}} \operatorname {hypergeom}\left (\left [\sqrt {n}\, \sqrt {n +1}, \sqrt {n}\, \sqrt {n +1}+1\right ], \left [1+2 \sqrt {n}\, \sqrt {n +1}\right ], -\frac {1}{r}\right ) \]

✓ Mathematica. Time used: 0.784 (sec). Leaf size: 92

ode=(r+r^2)*D[R[r],{r,2}]+r*D[R[r],r]-n*(n+1)*R[r]==0; 
ic={}; 
DSolve[{ode,ic},R[r],r,IncludeSingularSolutions->True]
 

\begin{align*} R(r)&\to c_2 G_{2,2}^{2,0}\left (-r\left | \begin {array}{c} 1-\sqrt {n} \sqrt {n+1},\sqrt {n} \sqrt {n+1}+1 \\ 0,1 \\ \end {array} \right .\right )+c_1 r \operatorname {Hypergeometric2F1}\left (1-\sqrt {n} \sqrt {n+1},\sqrt {n} \sqrt {n+1}+1,2,-r\right ) \end{align*}

✗ Sympy

from sympy import * 
r = symbols("r") 
n = symbols("n") 
R = Function("R") 
ode = Eq(-n*(n + 1)*R(r) + r*Derivative(R(r), r) + (r**2 + r)*Derivative(R(r), (r, 2)),0) 
ics = {} 
dsolve(ode,func=R(r),ics=ics)
 

False