2.1.64 Problem 64

Internal problem ID [8776]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 64
Date solved : Thursday, March 13, 2025 at 06:09:37 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

Solve

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 106

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

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

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
-> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
trying symmetries linear in x and y(x) 
trying differential order: 2; exact nonlinear 
trying 2nd order, integrating factor of the form mu(y) 
trying 2nd order, integrating factor of the form mu(x,y) 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the general case 
trying 2nd order, integrating factor of the form mu(y,y) 
-> Calling odsolve with the ODE`, -(_y1^3-4)*y(x)/_y1^3+2*((diff(y(x), x))*x+_y1)/_y1^3, y(x)`   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   <- 1st order linear successful 
trying differential order: 2; mu polynomial in y 
trying 2nd order, integrating factor of the form mu(x,y) 
differential order: 2; looking for linear symmetries 
differential order: 2; found: 1 linear symmetries. Trying reduction of order 
`, `2nd order, trying reduction of order with given symmetries:`[x, y]
 

✗ 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))