58.2.5 problem 5

Internal problem ID [9128]
Book : Second order enumerated odes
Section : section 2
Problem number : 5
Date solved : Sunday, March 30, 2025 at 02:17:08 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 61

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\frac {\sqrt {3}\, \int \tan \left (\operatorname {RootOf}\left (-\sqrt {3}\, \ln \left (\cos \left (\textit {\_Z} \right )^{2}\right )-2 \sqrt {3}\, \ln \left (\tan \left (\textit {\_Z} \right )+\sqrt {3}\right )+6 \sqrt {3}\, c_1 +6 \sqrt {3}\, x +6 \textit {\_Z} \right )\right )d x}{2}+c_2 +\frac {x}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.503 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{(K[1]+1) \left (K[1]^2-K[1]+1\right )}dK[1]\&\right ][c_1-K[2]]dK[2]\right ) \]

✗ Sympy

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

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