54.7.176 problem 1796 (book 6.205)

Internal problem ID [13025]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1796 (book 6.205)
Date solved : Friday, October 03, 2025 at 03:58:21 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{2} y^{\prime \prime }-a&=0 \end{align*}

✓ Maple. Time used: 0.180 (sec). Leaf size: 553

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 116

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

\[ \text {Solve}\left [-\frac {a \arctan \left (\frac {\sqrt {2} \sqrt {c_1} \left (\frac {y(x)}{x}+\frac {a}{2 c_1}\right )}{\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}\right )}{2 \sqrt {2} c_1{}^{3/2}}-\frac {\sqrt {-\frac {2 a y(x)}{x}-\frac {2 c_1 y(x)^2}{x^2}}}{2 c_1}-\frac {1}{x}-c_2=0,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : solve: Cannot solve -a + x*y(x)**2*Derivative(y(x), (x, 2))