2.1.63 Problem 63

Internal problem ID [8775]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 63
Date solved : Wednesday, March 05, 2025 at 06:47:43 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

Solve

\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 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*x-1)*y(x)/(_y1^3*x)+(1/3)*(3*(diff(y(x), x))*x+2*_y1)/(x*_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]
 

✗ 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))