79.6.5 problem (e)

Internal problem ID [21101]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter 1. First order equations: Some integrable cases. Excercises VIII at page 51
Problem number : (e)
Date solved : Thursday, October 02, 2025 at 07:08:20 PM
CAS classification : [_dAlembert]

\begin{align*} x&=y \left (y^{\prime }+\frac {1}{y^{\prime }}\right )+{y^{\prime }}^{5} \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 110

ode:=x = y(x)*(diff(y(x),x)+1/diff(y(x),x))+diff(y(x),x)^5; 
dsolve(ode,y(x), singsol=all);
 

\[ \left [x \left (\textit {\_T} \right ) = \frac {{\mathrm e}^{-\frac {1}{2 \textit {\_T}^{2}}} \left (\textit {\_T}^{2}+1\right ) \left (-2 \int \frac {\textit {\_T}^{4} \left (2 \textit {\_T}^{2}+3\right ) {\mathrm e}^{\frac {1}{2 \textit {\_T}^{2}}}}{\left (\textit {\_T}^{2}+1\right )^{2}}d \textit {\_T} +c_1 \right )}{\textit {\_T}^{2}}, y \left (\textit {\_T} \right ) = \frac {{\mathrm e}^{-\frac {1}{2 \textit {\_T}^{2}}} \textit {\_T} \left (1+\frac {1}{\textit {\_T}^{2}}\right ) \left (-2 \int \frac {\textit {\_T}^{4} \left (2 \textit {\_T}^{2}+3\right ) {\mathrm e}^{\frac {1}{2 \textit {\_T}^{2}}}}{\left (\textit {\_T}^{2}+1\right )^{2}}d \textit {\_T} +c_1 \right )-\textit {\_T}^{6}}{\textit {\_T}^{2}+1}\right ] \]

✓ Mathematica. Time used: 2.788 (sec). Leaf size: 7957

ode=x==y[x]*(D[y[x],x]+1/D[y[x],x])+D[y[x],x]^5; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - (Derivative(y(x), x) + 1/Derivative(y(x), x))*y(x) - Derivative(y(x), x)**5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x - (Derivative(y(x), x) + 1/Derivative(y(x), x))*y(x) - Derivative(y(x), x)**5 cannot be solved by the lie group method