54.7.75 problem 1684 (book 6.93)

Internal problem ID [12924]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1684 (book 6.93)
Date solved : Friday, October 03, 2025 at 03:50:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 23

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

\[ y = -\frac {\ln \left (\frac {a \left (c_1 x -c_2 \right )}{x}\right ) x}{a} \]

✓ Mathematica. Time used: 1.854 (sec). Leaf size: 25

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

\begin{align*} y(x)&\to -\frac {x \log \left (-\frac {a (c_2 x+c_1)}{x}\right )}{a} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - y(x)/x - sqrt(a*x**5*Derivative(y(x), (x, 2)))/(a*x**2) cannot be solved by the factorable group method