2.1.22 Problem 22

Internal problem ID [10381]
Book : Second order enumerated odes
Section : section 1
Problem number : 22
Date solved : Thursday, November 27, 2025 at 10:39:37 AM
CAS classification : [[_2nd_order, _missing_x]]

\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

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

\[ \text {No solution found} \]

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 e\ 
ach 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 throug\ 
h 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 th\ 
e 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 thro\ 
ugh one integrating factor of the form G(x,y)/(1+H(x,y)*y)^2 
   solving 2nd order ODE of high degree, Lie methods 
   2nd order, trying reduction of order with given symmetries: 
[1, 0] 
      -> 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 3 *** 
      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 Lie symmetry methods, 1st order --- 
         -> Computing symmetries using: way = 2 
         -> Computing symmetries using: way = 3 
         -> Computing symmetries using: way = 4 
         -> Computing symmetries using: way = 5 
      trying symmetry patterns for 1st order ODEs 
      -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
      -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
      -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 
      -> trying a symmetry pattern of the form [F(x),G(x)] 
      -> trying a symmetry pattern of the form [F(y),G(y)] 
      -> trying a symmetry pattern of the form [F(x)+G(y), 0] 
      -> trying a symmetry pattern of the form [0, F(x)+G(y)] 
      -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
      -> trying a symmetry pattern of conformal type
 

✗ Mathematica

ode=(D[y[x],{x,2}])^2+D[y[x],x]+y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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