60.7.162 problem 1780 (book 6.189)

Internal problem ID [11712]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1780 (book 6.189)
Date solved : Sunday, March 30, 2025 at 08:44:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 110

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

\[ \ln \left (x \right )-\frac {\int _{}^{\frac {y}{x}}\frac {\textit {\_g}^{2} \left (\left (\left (\frac {a}{\textit {\_g}^{3}}\right )^{{1}/{3}}-2\right ) \sqrt {3}+3 \left (\frac {a}{\textit {\_g}^{3}}\right )^{{1}/{3}} \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {3}-\ln \left (\frac {1}{\sqrt {3}\, \sin \left (2 \textit {\_Z} \right )+2+\cos \left (2 \textit {\_Z} \right )}\right )-6 c_1 -6 \int \frac {\left (\frac {a}{\textit {\_g}^{3}}\right )^{{2}/{3}} \textit {\_g}^{2}}{\textit {\_g}^{3}+a}d \textit {\_g} \right )\right )\right )}{\textit {\_g}^{3}+a}d \textit {\_g} \sqrt {3}}{6}-c_2 = 0 \]

✗ Mathematica

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

Not solved

✗ 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)) + y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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