42.4.17 problem Problem 3.24

Internal problem ID [8845]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page 218
Problem number : Problem 3.24
Date solved : Friday, October 03, 2025 at 02:13:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (1+x \right ) \eta ^{\prime }+\left (k +1\right ) \eta &=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 95

ode:=(-x^2+1)*diff(diff(eta(x),x),x)-(1+x)*diff(eta(x),x)+(1+k)*eta(x) = 0; 
dsolve(ode,eta(x), singsol=all);
 

\[ \eta = c_1 \left (1+x \right )^{\sqrt {k +1}} \operatorname {hypergeom}\left (\left [-\sqrt {k +1}, -\sqrt {k +1}+1\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{1+x}\right )+c_2 \left (1+x \right )^{-\sqrt {k +1}} \operatorname {hypergeom}\left (\left [\sqrt {k +1}, \sqrt {k +1}+1\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{1+x}\right ) \]

✓ Mathematica. Time used: 0.26 (sec). Leaf size: 77

ode=(1-x^2)*D[z[x],{x,2}]-(1+x)*D[z[x],x]+(k+1)*z[x]==0; 
ic={}; 
DSolve[{ode,ic},z[x],x,IncludeSingularSolutions->True]
 

\begin{align*} z(x)&\to c_2 G_{2,2}^{2,0}\left (\frac {1-x}{2}| \begin {array}{c} 1-\sqrt {k+1},\sqrt {k+1}+1 \\ 0,0 \\ \end {array} \right )+c_1 \operatorname {Hypergeometric2F1}\left (-\sqrt {k+1},\sqrt {k+1},1,\frac {1-x}{2}\right ) \end{align*}

✗ Sympy

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

False