44.11.16 problem 5(a)

Internal problem ID [9286]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.3. THE METHOD OF VARIATION OF PARAMETERS. Page 71
Problem number : 5(a)
Date solved : Tuesday, September 30, 2025 at 06:16:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

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

\[ y = c_2 x +c_1 \,x^{2}+c_1 +\frac {1}{2}+\frac {1}{6} x^{4} \]

✓ Mathematica. Time used: 0.264 (sec). Leaf size: 192

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

\begin{align*} y(x)&\to \sqrt {x^2-1} \exp \left (\int _1^x\frac {K[1]+2}{K[1]^2-1}dK[1]\right ) \left (\int _1^x-\exp \left (\int _1^{K[3]}\frac {K[1]+2}{K[1]^2-1}dK[1]\right ) \sqrt {K[3]^2-1} \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2}{K[1]^2-1}dK[1]\right )dK[2]dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]+2}{K[1]^2-1}dK[1]\right )dK[2] \left (\int _1^x\exp \left (\int _1^{K[4]}\frac {K[1]+2}{K[1]^2-1}dK[1]\right ) \sqrt {K[4]^2-1}dK[4]+c_2\right )+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*(-x**2 + Derivative(y(x), (x, 2)) + 2) + 2*y(x) - Derivative(y(x), (x, 2)) - 1)/(2*x) cannot be solved by the factorable group method