54.3.344 problem 1361

Internal problem ID [12639]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1361
Date solved : Wednesday, October 01, 2025 at 02:18:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x) = 2*x/(x^2-1)*diff(y(x),x)-(a*(a+1)-a*x^2*(a+3))/x^2/(x^2-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{-a}+c_2 \,x^{a +1} \left (2 x^{2} a +x^{2}-2 a -3\right ) \]

✓ Mathematica. Time used: 0.509 (sec). Leaf size: 67

ode=D[y[x],{x,2}] == -(((a*(1 + a) - a*(3 + a)*x^2)*y[x])/(x^2*(-1 + x^2))) + (2*x*D[y[x],x])/(-1 + x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 \exp \left (\int \frac {2 a^2 \left (x^2-1\right )+a \left (7 x^2-5\right )+3 \left (x^2-1\right )}{x \left (2 a \left (x^2-1\right )+x^2-3\right )} \, dx\right )+c_1 x^{-a} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x)/(x**2 - 1) + Derivative(y(x), (x, 2)) + (-a*x**2*(a + 3) + a*(a + 1))*y(x)/(x**2*(x**2 - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None