54.3.346 problem 1363

Internal problem ID [12641]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1363
Date solved : Friday, October 03, 2025 at 03:45:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (a \,x^{2}+a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {b y}{x^{2}} \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 253

ode:=diff(diff(y(x),x),x) = -1/x*(a*x^2+a-2)/(x^2-1)*diff(y(x),x)-b/x^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{-\frac {1}{2}+\frac {a}{2}} \left (-x^{2}+1\right )^{\frac {1}{2}+\frac {a}{2}} \left (x^{2}-1\right )^{-a} \left (c_1 \,x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right ) \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, -\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right )+\frac {\pi c_2 \,x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \sqrt {a^{2}-2 a -4 b +1}\, \csc \left (\frac {\pi \sqrt {a^{2}-2 a -4 b +1}}{2}\right ) \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, \frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right )}{2 \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right )}\right ) \]

✓ Mathematica. Time used: 0.437 (sec). Leaf size: 212

ode=D[y[x],{x,2}] == -((b*y[x])/x^2) - ((-2 + a + a*x^2)*D[y[x],x])/(x*(-1 + x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -(-1)^{\frac {1}{4} \left (-\sqrt {a^2-2 a-4 b+1}+a+3\right )} x^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}+a-1\right )} \left (c_1 \operatorname {Hypergeometric2F1}\left (\frac {a-1}{2},\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 b+1}-1\right ),1-\frac {1}{2} \sqrt {a^2-2 a-4 b+1},x^2\right )+c_2 i^{\sqrt {a^2-2 a-4 b+1}} x^{\sqrt {a^2-2 a-4 b+1}} \operatorname {Hypergeometric2F1}\left (\frac {a-1}{2},\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 b+1}-1\right ),\frac {1}{2} \left (\sqrt {a^2-2 a-4 b+1}+2\right ),x^2\right )\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None