69.21.8 problem 703

Internal problem ID [18464]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 16. The method of isoclines for differential equations of the second order. Exercises page 158
Problem number : 703
Date solved : Thursday, October 02, 2025 at 03:12:40 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 31

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

\[ x = -i \operatorname {RootOf}\left (i \sqrt {2}\, c_1 t +i \sqrt {2}\, c_2 -\operatorname {erf}\left (\textit {\_Z} \right ) \sqrt {\pi }\right ) \sqrt {2} \]

✓ Mathematica. Time used: 0.944 (sec). Leaf size: 34

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

\begin{align*} x(t)&\to -i \sqrt {2} \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} c_1 (t+c_2)\right ) \end{align*}

✗ Sympy

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

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