60.7.142 problem 1759 (book 6.168)

Internal problem ID [11692]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1759 (book 6.168)
Date solved : Sunday, March 30, 2025 at 08:42:47 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (a y+b \right ) y^{\prime \prime }+c {y^{\prime }}^{2}&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 73

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

\begin{align*} y &= -\frac {b}{a} \\ y &= \frac {\left (\frac {1}{\left (c +a \right ) \left (c_1 x +c_2 \right )}\right )^{\frac {c}{c +a}} \left (-\left (\frac {1}{\left (c +a \right ) \left (c_1 x +c_2 \right )}\right )^{-\frac {c}{c +a}} b +\left (c +a \right ) \left (c_1 x +c_2 \right )\right )}{a} \\ \end{align*}

✓ Mathematica. Time used: 8.823 (sec). Leaf size: 31

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

\[ y(x)\to \frac {-b+(c_1 (a+c) (x+c_2)){}^{\frac {a}{a+c}}}{a} \]

✗ Sympy

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

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