2.1.63 Problem 63

Internal problem ID [8775]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 63
Date solved : Friday, February 21, 2025 at 08:30:25 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _with_linear_symmetries]]

Solve

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

Maple step by step solution

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]
 

Maple dsolve solution

Solving time : 0.086 (sec)
Leaf size : maple_leaf_size

dsolve(y(x)*diff(diff(y(x),x),x) = x,y(x),singsol=all)
 

\[ \text {No solution found} \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

Not solved

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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))