66.2.27 problem Problem 36

Internal problem ID [13855]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 36
Date solved : Monday, March 31, 2025 at 08:15:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-6 y&=1 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 52

ode:=(x^2-1)*diff(diff(y(x),x),x)-6*y(x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {1}{6}+\frac {3 \left (x^{3}-x \right ) c_1 \ln \left (x -1\right )}{4}+\frac {3 c_1 \left (-x^{3}+x \right ) \ln \left (x +1\right )}{4}+c_2 \,x^{3}+\frac {3 c_1 \,x^{2}}{2}-c_2 x -c_1 \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 67

ode=(x^2-1)*D[y[x],{x,2}]-6*y[x]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{12} \left (-9 c_2 x \left (x^2-1\right ) \log (1-x)+9 c_2 x \left (x^2-1\right ) \log (x+1)+2 \left (6 c_1 x^3-9 c_2 x^2-6 c_1 x-1+6 c_2\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 - 1)*Derivative(y(x), (x, 2)) - 6*y(x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve (x**2 - 1)*Derivative(y(x), (x, 2)) - 6*y(x) - 1