56.1.64 problem 64

Internal problem ID [8776]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 64
Date solved : Sunday, March 30, 2025 at 01:32:58 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 106

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

\[ y = \operatorname {RootOf}\left (\ln \left (x \right )+2^{{1}/{3}} \int _{}^{\textit {\_Z}}\frac {1}{2^{{1}/{3}} \textit {\_f} +2 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{{2}/{3}}}{2 \textit {\_f}}\right ) c_1 \textit {\_Z} +\textit {\_Z} \operatorname {AiryAi}\left (\frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{{2}/{3}}}{2 \textit {\_f}}\right )+\operatorname {AiryBi}\left (1, \frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{{2}/{3}}}{2 \textit {\_f}}\right ) c_1 +\operatorname {AiryAi}\left (1, \frac {2 \textit {\_Z}^{2} \textit {\_f} +2^{{2}/{3}}}{2 \textit {\_f}}\right )\right )}d \textit {\_f} -c_2 \right ) x \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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