54.7.148 problem 1765 (book 6.174)

Internal problem ID [12997]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1765 (book 6.174)
Date solved : Friday, October 03, 2025 at 03:58:14 AM
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 (y+1\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.114 (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: 0.03 (sec). Leaf size: 31

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

\[ \text {Solve}\left [\log (x)+\int _1^{y(x)}\frac {2}{-2 c_1 K[1]^2-1}dK[1]=c_2,y(x)\right ] \]

✗ 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