60.7.159 problem 1776 (book 6.185)

Internal problem ID [11709]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1776 (book 6.185)
Date solved : Sunday, March 30, 2025 at 08:43:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x \left (x +1\right )^{2} y y^{\prime \prime }-x \left (x +1\right )^{2} {y^{\prime }}^{2}+2 \left (x +1\right )^{2} y y^{\prime }-a \left (x +2\right ) y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 31

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

\begin{align*} y &= 0 \\ y &= \frac {\left (x +1\right )^{a} {\mathrm e}^{\frac {c_2 +\left (-x -1\right ) a}{x}}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 10.824 (sec). Leaf size: 47

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

\[ y(x)\to c_2 \exp \left (\int _1^x\frac {c_1+\int _1^{K[2]}\frac {a K[1] (K[1]+2)}{(K[1]+1)^2}dK[1]}{K[2]^2}dK[2]\right ) \]

✗ Sympy

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

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