58.2.9 problem 10

Internal problem ID [9132]
Book : Second order enumerated odes
Section : section 2
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 07:33:46 AM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 37

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

\[ y = -i \operatorname {RootOf}\left (i \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right ) c_{1} +i \sqrt {2}\, c_{2} -\operatorname {erf}\left (\textit {\_Z} \right ) \sqrt {\pi }\right ) \sqrt {2} \]

✓ Mathematica. Time used: 1.341 (sec). Leaf size: 44

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

\[ y(x)\to -i \sqrt {2} \text {erf}^{-1}\left (i \left (\sqrt {\frac {2}{\pi }} c_2-c_1 \text {erf}\left (\frac {x}{\sqrt {2}}\right )\right )\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE -(-x + sqrt(x**2 - 4*y(x)*Derivative(y(x), (x, 2))))/(2*y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method