1.22 problem 22

Internal problem ID [7411]
Internal file name [OUTPUT/6378_Sunday_June_05_2022_04_42_21_PM_1760656/index.tex]

Book: Second order enumerated odes
Section: section 1
Problem number: 22.
ODE order: 2.
ODE degree: 2.

The type(s) of ODE detected by this program : "unknown"

Maple gives the following as the ode type

[[_2nd_order, _missing_x]]

Unable to solve or complete the solution.

\[ \boxed {{y^{\prime \prime }}^{2}+y^{\prime }+y=0} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(1\) is missing which is not the case here.

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]
 

✗ Solution by Maple

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

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[(y''[x])^2+y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

Not solved