54.7.77 problem 1686 (book 6.95)

Internal problem ID [12926]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1686 (book 6.95)
Date solved : Wednesday, October 01, 2025 at 02:46:17 AM
CAS classification : [NONE]

\begin{align*} 2 \left (-x^{k}+4 x^{3}\right ) \left (y^{\prime \prime }+y y^{\prime }-y^{3}\right )-\left (k \,x^{k -1}-12 x^{2}\right ) \left (3 y^{\prime }+y^{2}\right )+y a x +b&=0 \end{align*}

✗ Maple

ode:=2*(-x^k+4*x^3)*(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3)-(k*x^(k-1)-12*x^2)*(3*diff(y(x),x)+y(x)^2)+y(x)*a*x+b = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=b + a*x*y[x] - (-12*x^2 + k*x^(-1 + k))*(y[x]^2 + 3*D[y[x],x]) + 2*(4*x^3 - x^k)*(-y[x]^3 + y[x]*D[y[x],x] + 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") 
y = Function("y") 
ode = Eq(a*x*y(x) + b + (8*x**3 - 2*x**k)*(-y(x)**3 + y(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2))) - (k*x**(k - 1) - 12*x**2)*(y(x)**2 + 3*Derivative(y(x), x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a*x*y(x) + b - k*x**(k - 1)*y(x)**2 - 8*x**3*y(x)**3 + 8*x**3*Derivative(y(x), (x, 2)) + 12*x**2*y(x)**2 + 2*x**k*y(x)**3 - 2*x**k*Derivative(y(x), (x, 2)))/(3*k*x**(k - 1) - 8*x**3*y(x) - 36*x**2 + 2*x**k*y(x)) cannot be solved by the factorable group method