75.2.5 problem 5

Internal problem ID [19899]
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 : 5
Date solved : Thursday, October 02, 2025 at 05:00:34 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(sqrt((y(x)**2 + 1)*(-2*y(x)**3 + y(x)**2 + 2*y(x)*Derivative(y(x), (x, 2)) + 1)) - y(x)**2 - 1)/(2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method