60.7.103 problem 1715 (book 6.124)

Internal problem ID [11653]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1715 (book 6.124)
Date solved : Sunday, March 30, 2025 at 08:33:12 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime } y-3 {y^{\prime }}^{2}+3 y y^{\prime }-y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 72

ode:=diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2+3*y(x)*diff(y(x),x)-y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= -\frac {\sqrt {2}\, \sqrt {\left ({\mathrm e}^{x} c_1 -c_2 \right ) {\mathrm e}^{2 x}}}{2 \,{\mathrm e}^{x} c_1 -2 c_2} \\ y &= \frac {\sqrt {2}\, \sqrt {\left ({\mathrm e}^{x} c_1 -c_2 \right ) {\mathrm e}^{2 x}}}{2 \,{\mathrm e}^{x} c_1 -2 c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.464 (sec). Leaf size: 47

ode=-y[x]^2 + 3*y[x]*D[y[x],x] - 3*D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) (2 K[1]-1)}dK[1]\&\right ][c_1+K[2]]dK[2]\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x)**2 + 3*y(x)*Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(3)*sqrt((-y(x) + 4*Derivative(y(x), (x, 2)))*y(x))/6 - y(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method