32.10.24 problem Exercise 35.23(c), page 504

Internal problem ID [6018]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.23(c), page 504
Date solved : Thursday, March 13, 2025 at 05:44:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.044 (sec). Leaf size: 22

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

\begin{align*} y &= 0 \\ y &= c_1 \tanh \left (\frac {\ln \left (x \right )-c_2}{2 c_1}\right ) \\ \end{align*}

✓ Mathematica. Time used: 22.975 (sec). Leaf size: 52

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

\begin{align*} y(x)\to \frac {\tan \left (\frac {\sqrt {c_1} (\log (x)-c_2)}{\sqrt {2}}\right )}{\sqrt {2} \sqrt {c_1}} \\ y(x)\to \frac {1}{2} (\log (x)-c_2) \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(8*x**2*y(x)*Derivative(y(x), (x, 2)) + y(x)**2 + 2*y(x) + 1) + y(x) + 1)/(4*x) cannot be solved by the factorable group method