60.7.109 problem 1722 (book 6.131)

Internal problem ID [11659]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1722 (book 6.131)
Date solved : Sunday, March 30, 2025 at 08:36:42 PM
CAS classification : [NONE]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 48.973 (sec). Leaf size: 46

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

\begin{align*} y(x)\to -\frac {(a+2) (x+c_1){}^a}{a \int _1^xf(K[5]) (c_1+K[5]){}^adK[5]+c_2} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a**2*f(x)*y(x)**2/2 - a*f(x)*y(x)**2 + sqrt(a*(a**3*f(x)**2*y(x)**3 - 4*a**3*y(x)**2*Derivative(f(x), x) + 4*a**3*Derivative(y(x), (x, 2)) + 8*a**2*f(x)**2*y(x)**3 - 4*a**2*y(x)**2*Derivative(f(x), x) + 12*a**2*Derivative(y(x), (x, 2)) + 8*a*y(x)**2*Derivative(f(x), x) - 16*Derivative(y(x), (x, 2)))*y(x))/2)/(a**2 + a - 2) cannot be solved by the factorable group method