54.7.143 problem 1760 (book 6.169)

Internal problem ID [12992]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1760 (book 6.169)
Date solved : Wednesday, October 01, 2025 at 02:55:49 AM
CAS classification : [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y y^{\prime \prime }+x {y^{\prime }}^{2}-y y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 35

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

\begin{align*} y &= 0 \\ y &= \sqrt {c_1 \,x^{2}+2 c_2} \\ y &= -\sqrt {c_1 \,x^{2}+2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to c_2 \sqrt {x^2+c_1} \end{align*}

✗ Sympy

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

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