77.38.7 problem 7

Internal problem ID [20750]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise VIII (C) at page 133
Problem number : 7
Date solved : Thursday, October 02, 2025 at 06:22:47 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: 0.002 (sec). Leaf size: 24

ode:=(x^2+1)^2*diff(diff(y(x),x),x)+2*x*(x^2+1)*diff(y(x),x)+4*y(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: 0.021 (sec). Leaf size: 22

ode=(1+x^2)^2*D[y[x],{x,2}]+(2*x)*(1+x^2)*D[y[x],x]+4*y[x]==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