59.5.9 problem 10.4 (ii)

Internal problem ID [15029]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 10, Two tricks for nonlinear equations. Exercises page 97
Problem number : 10.4 (ii)
Date solved : Thursday, October 02, 2025 at 10:02:00 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} x^{\prime }&=\frac {x^{2}+t \sqrt {t^{2}+x^{2}}}{x t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(x(t),t) = (x(t)^2+t*(t^2+x(t)^2)^(1/2))/t/x(t); 
dsolve(ode,x(t), singsol=all);
 

\[ \frac {\ln \left (t \right ) t -c_1 t -\sqrt {t^{2}+x^{2}}}{t} = 0 \]

✓ Mathematica. Time used: 0.17 (sec). Leaf size: 54

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

\begin{align*} x(t)&\to -t \sqrt {\log ^2(t)+2 c_1 \log (t)-1+c_1{}^2}\\ x(t)&\to t \sqrt {\log ^2(t)+2 c_1 \log (t)-1+c_1{}^2} \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: 2*t**2 < 2*C1*t**2