54.7.36 problem 1631 (6.41)

Internal problem ID [12885]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1631 (6.41)
Date solved : Friday, October 03, 2025 at 03:50:38 AM
CAS classification : [[_2nd_order, _with_potential_symmetries]]

\begin{align*} y^{\prime \prime }-\left (3 y+f \left (x \right )\right ) y^{\prime }+y^{3}+y^{2} f \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 37

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

\[ y = \frac {-c_1 \int {\mathrm e}^{\int f \left (x \right )d x}d x -c_2}{c_1 \int \int {\mathrm e}^{\int f \left (x \right )d x}d x d x +c_2 x +1} \]

✓ Mathematica. Time used: 38.011 (sec). Leaf size: 81

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

\begin{align*} y(x)&\to -\frac {\int _1^x\exp \left (\int _1^{K[2]}f(K[1])dK[1]\right ) c_1dK[2]+c_2}{\int _1^x\int _1^{K[5]}\exp \left (\int _1^{K[4]}f(K[3])dK[3]\right ) c_1dK[4]dK[5]+c_2 x+1}\\ y(x)&\to -\frac {1}{x} \end{align*}

✗ Sympy

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

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