58.2.6 problem 6

Internal problem ID [9129]
Book : Second order enumerated odes
Section : section 2
Problem number : 6
Date solved : Sunday, March 30, 2025 at 02:17:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 169

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

\begin{align*} \frac {\left (-2-2 n \right ) \int _{}^{y}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_1 \right ) \left (1+n \right )^{2}\right )}^{{1}/{3}}}d \textit {\_a} -\left (1+i \sqrt {3}\right ) \left (x +c_2 \right )}{1+i \sqrt {3}} &= 0 \\ -\frac {2 i \left (1+n \right ) \int _{}^{y}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_1 \right ) \left (1+n \right )^{2}\right )}^{{1}/{3}}}d \textit {\_a} +\left (x +c_2 \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \int _{}^{y}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_1 \right ) \left (1+n \right )^{2}\right )}^{{1}/{3}}}d \textit {\_a} n +\int _{}^{y}\frac {1}{{\left (-\left (3 \textit {\_a}^{1+n}-c_1 \right ) \left (1+n \right )^{2}\right )}^{{1}/{3}}}d \textit {\_a} -c_2 -x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 2.216 (sec). Leaf size: 910

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(y(x)**n + Derivative(y(x), x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE y(x)**n/Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method