54.7.25 problem 1615 (6.25)

Internal problem ID [12874]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1615 (6.25)
Date solved : Wednesday, October 01, 2025 at 02:28:29 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-\frac {\left (3 n +4\right ) y^{\prime }}{n}-\frac {2 \left (n +1\right ) \left (n +2\right ) y \left (y^{\frac {n}{n +1}}-1\right )}{n^{2}}&=0 \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x)-(3*n+4)/n*diff(y(x),x)-2*(n+1)*(n+2)/n^2*y(x)*(y(x)^(n/(n+1))-1) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=(-2*(1 + n)*(2 + n)*y[x]*(-1 + y[x]^(n/(1 + n))))/n^2 - ((4 + 3*n)*D[y[x],x])/n + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (3*n + 4)*Derivative(y(x), x)/n - (n + 2)*(2*n + 2)*(y(x)**(n/(n + 1)) - 1)*y(x)/n**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (2*n**2*y(x) - 2*n**2*y(x)**(n/(n + 1) + 1) + n**2*Derivative(y(x), (x, 2)) + 6*n*y(x) - 6*n*y(x)**(n/(n + 1) + 1) + 4*y(x) - 4*y(x)**(n/(n + 1) + 1))/(n*(3*n + 4)) cannot be solved by the factorable group method