54.7.96 problem 1708 (book 6.117)

Internal problem ID [12945]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1708 (book 6.117)
Date solved : Wednesday, October 01, 2025 at 02:46:38 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 28

ode=b*y[x]^2 + a*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]
 

\begin{align*} y(x)&\to c_2 e^{-\frac {b x+c_1 e^{-a x}}{a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*y(x)*Derivative(y(x), x) + b*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 -a*y(x)/2 - sqrt((a**2*y(x) + 4*b*y(x) + 4*Derivative(y(x), (x, 2)))*y(x))/2 + Derivative(y(x), x) cannot be solved by the factorable group method