54.7.102 problem 1714 (book 6.123)

Internal problem ID [12951]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1714 (book 6.123)
Date solved : Wednesday, October 01, 2025 at 02:48:12 AM
CAS classification : [[_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 32.81 (sec). Leaf size: 312

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

\begin{align*} y(x)&\to -\frac {\exp \left (\int _1^x\frac {f''(K[3]) y(K[3])^3-\left (c_1{}^2+2 \int _1^{K[3]}\left (y(K[1]) f''(K[1])-\frac {g''(K[1])}{y(K[1])}+f(K[1]) y''(K[1])+\frac {g(K[1]) y''(K[1])}{y(K[1])^2}\right )dK[1] c_1+\int _1^{K[3]}\left (y(K[1]) f''(K[1])-\frac {g''(K[1])}{y(K[1])}+f(K[1]) y''(K[1])+\frac {g(K[1]) y''(K[1])}{y(K[1])^2}\right )dK[1]{}^2-f(K[3]) y''(K[3])\right ) y(K[3])^2-g''(K[3]) y(K[3])+g(K[3]) y''(K[3])}{y(K[3])^2 \left (c_1+\int _1^{K[3]}\left (y(K[1]) f''(K[1])-\frac {g''(K[1])}{y(K[1])}+f(K[1]) y''(K[1])+\frac {g(K[1]) y''(K[1])}{y(K[1])^2}\right )dK[1]\right )}dK[3]-c_2\right )}{\int _1^x\left (y(K[1]) f''(K[1])-\frac {g''(K[1])}{y(K[1])}+f(K[1]) y''(K[1])+\frac {g(K[1]) y''(K[1])}{y(K[1])^2}\right )dK[1]+c_1}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

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