60.7.46 problem 1648 (6.58)

Internal problem ID [11596]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1648 (6.58)
Date solved : Sunday, March 30, 2025 at 08:30:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-k \,x^{a} y^{b} {y^{\prime }}^{r}&=0 \end{align*}

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -(Derivative(y(x), (x, 2))/(k*x**a*y(x)**b))**(1/r) + Derivative(y(x), x) cannot be solved by the factorable group method