90.3.18 problem 18

Internal problem ID [25082]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 41
Problem number : 18
Date solved : Thursday, October 02, 2025 at 11:49:36 PM
CAS classification : [_quadrature]

\begin{align*} y y^{\prime }&=1+y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

\begin{align*} y &= \sqrt {{\mathrm e}^{2 t} c_1 -1} \\ y &= -\sqrt {{\mathrm e}^{2 t} c_1 -1} \\ \end{align*}

✓ Mathematica. Time used: 1.824 (sec). Leaf size: 53

ode=y[t]*D[y[t],{t,1}] ==y[t]^2+1; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\sqrt {-1+e^{2 (t+c_1)}}\\ y(t)&\to \sqrt {-1+e^{2 (t+c_1)}}\\ y(t)&\to -i\\ y(t)&\to i \end{align*}

✓ Sympy. Time used: 0.341 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)**2 + y(t)*Derivative(y(t), t) - 1,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = - \sqrt {C_{1} e^{2 t} - 1}, \ y{\left (t \right )} = \sqrt {C_{1} e^{2 t} - 1}\right ] \]