30.5.13 problem 13

Internal problem ID [7512]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.6, Substitutions and Transformations. Exercises. page 76
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 04:40:59 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 28

ode:=diff(x(t),t) = (x(t)^2+t*(t^2+x(t)^2)^(1/2))/x(t)/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: 0.185 (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