81.2.10 problem 10

Internal problem ID [18547]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 10
Date solved : Monday, March 31, 2025 at 05:42:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

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

\[ v = \frac {c_1 x +c_2}{\sqrt {x^{2}+1}} \]

✓ Mathematica. Time used: 2.036 (sec). Leaf size: 22

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

\[ v(x)\to \frac {c_2 x+c_1}{\sqrt {x^2+1}} \]

✗ Sympy

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

False