62.30.3 problem Ex 3

Internal problem ID [12896]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 53. Change of dependent variable. Page 125
Problem number : Ex 3
Date solved : Monday, March 31, 2025 at 07:23:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

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

\[ y = c_1 x +\arctan \left (x \right ) x c_2 +c_2 \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 48

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

\[ y(x)\to \frac {1}{2} i (2 c_1 x-c_2 x \log (1-i x)+c_2 x \log (1+i x)+2 i c_2) \]

✗ Sympy

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

False