60.7.101 problem 1713 (book 6.122)

Internal problem ID [11651]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1713 (book 6.122)
Date solved : Sunday, March 30, 2025 at 08:33:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.061 (sec). Leaf size: 37

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

\[ y = c_2 \,{\mathrm e}^{c_1 \int {\mathrm e}^{\int f \left (x \right )d x}d x +\int {\mathrm e}^{\int f \left (x \right )d x} \int {\mathrm e}^{-\int f \left (x \right )d x} g \left (x \right )d x d x} \]

✓ Mathematica. Time used: 1.187 (sec). Leaf size: 61

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

\[ y(x)\to c_2 \exp \left (\int _1^x\exp \left (\int _1^{K[3]}f(K[1])dK[1]\right ) \left (c_1+\int _1^{K[3]}\exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) g(K[2])dK[2]\right )dK[3]\right ) \]

✗ Sympy

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

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