23.3.495 problem 501

Internal problem ID [6209]
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 : 501
Date solved : Tuesday, September 30, 2025 at 02:36:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to \frac {1}{2} c_2 \left (\left (x^2+1\right ) \arctan (x)-x\right )+c_1 \left (x^2+1\right ) \end{align*}

✗ Sympy

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

False