2.1.22 Problem 22

Internal problem ID [9093]
Book : Second order enumerated odes
Section : section 1
Problem number : 22
Date solved : Friday, February 21, 2025 at 09:10:36 PM
CAS classification : [[_2nd_order, _missing_x]]

Solve

\begin{align*} {y^{\prime \prime }}^{2}+y^{\prime }+y&=0 \end{align*}

Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(1\) is missing which is not the case here.

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
   *** Sublevel 2 *** 
   Methods for second order ODEs: 
   Successful isolation of d^2y/dx^2: 2 solutions were found. Trying to solve each resulting ODE. 
      *** Sublevel 3 *** 
      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 
      `, `-> Computing symmetries using: way = 3 
      `, `-> Computing symmetries using: way = exp_sym 
      -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-(-_b(_a)-_a)^(1/2) = 0, _b(_a)`         *** Sublevel 4 *** 
         Methods for first order ODEs: 
         --- Trying classification methods --- 
         trying homogeneous types: 
         trying exact 
         Looking for potential symmetries 
         trying an equivalence to an Abel ODE 
         trying 1st order ODE linearizable_by_differentiation 
      -> trying 2nd order, dynamical_symmetries, fully reducible to Abel through one integrating factor of the form G(x,y)/(1+H(x,y) 
      trying 2nd order, integrating factors of the form mu(x,y)/(y)^n, only the singular cases 
      trying differential order: 2; exact nonlinear 
      trying 2nd order, integrating factor of the form mu(x,y) 
      -> trying 2nd order, the S-function method 
         -> trying a change of variables {x -> y(x), y(x) -> x} and re-entering methods 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 general case 
      -> trying 2nd order, dynamical_symmetries, only a reduction of order through one integrating factor of the form G(x,y)/(1+H(x, 
   solving 2nd order ODE of high degree, Lie methods 
`, `2nd order, trying reduction of order with given symmetries:`[1, 0]
 

Maple dsolve solution

Solving time : 0.064 (sec)
Leaf size : maple_leaf_size

dsolve(diff(diff(y(x),x),x)^2+diff(y(x),x)+y(x) = 0,y(x),singsol=all)
 

\[ \text {No solution found} \]

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

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

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