2.1.63 Problem 63

Internal problem ID [10049]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 63
Date solved : Thursday, November 27, 2025 at 10:07:34 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

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 sing\ 
ular 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 gener\ 
al case 
trying 2nd order, integrating factor of the form mu(y,y) 
-> Calling odsolve with the ODE, -(_y1^3*x-1)/_y1^3/x*y(x)+1/3/x*(3*diff(y(x),x 
)*x+2*_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, 3/2*y] 
   -> Calling odsolve with the ODE, diff(y(x),x) = 3/2*y(x)/x, y(x) 
      *** Sublevel 2 *** 
      Methods for first order ODEs: 
      --- Trying classification methods --- 
      trying a quadrature 
      trying 1st order linear 
      <- 1st order linear successful 
   -> Computing symmetries using: way = 3 
   -> Computing symmetries using: way = exp_sym 
Try integration with the canonical coordinates of the symmetry [x, 3/2*y] 
-> Calling odsolve with the ODE, diff(_b(_a),_a) = 1/4*_b(_a)^2*(3*_b(_a)*_a^2-\ 
4*_b(_a)+8*_a)/_a, _b(_a), explicit 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying 1st order linear 
   trying Bernoulli 
   trying separable 
   trying inverse linear 
   trying homogeneous types: 
   trying Chini 
   differential order: 1; looking for linear symmetries 
   trying exact 
   trying Abel 
   Looking for potential symmetries 
   Looking for potential symmetries 
   Looking for potential symmetries 
   trying inverse_Riccati 
   trying an equivalence to an Abel ODE 
   differential order: 1; trying a linearization to 2nd order 
   --- trying a change of variables {x -> y(x), y(x) -> x} 
   differential order: 1; trying a linearization to 2nd order 
   trying 1st order ODE linearizable_by_differentiation 
-> trying 2nd order, the S-function method 
   -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering metho\ 
ds for the S-function 
   -> trying 2nd order, the S-function method 
   -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
   -> trying 2nd order, No Point Symmetries Class V 
trying 2nd order, integrating factor of the form mu(x,y)/(y)^n, only the gener\ 
al case 
-> trying 2nd order, dynamical_symmetries, only a reduction of order through on\ 
e integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
--- Trying Lie symmetry methods, 2nd order --- 
   -> Computing symmetries using: way = 3 
[2/3*x, y] 
   <- successful computation of symmetries.
 

✗ Mathematica

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

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