23.3.546 problem 552

Internal problem ID [6260]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 552
Date solved : Tuesday, September 30, 2025 at 02:39:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 4 y+2 x \left (x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right )^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 24

ode:=4*y(x)+2*x*(x^2+1)*diff(y(x),x)+(x^2+1)^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 22

ode=4*y[x] + 2*x*(1 + x^2)*D[y[x],x] + (1 + x^2)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos (2 \arctan (x))+c_2 \sin (2 \arctan (x)) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(x**2 + 1)*Derivative(y(x), x) + (x**2 + 1)**2*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False