90.3.31 problem 31

Internal problem ID [25095]
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 : 31
Date solved : Thursday, October 02, 2025 at 11:50:03 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (1\right )&=\sqrt {3} \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 11

ode:=diff(y(t),t) = (y(t)^2+1)/t; 
ic:=[y(1) = 3^(1/2)]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \tan \left (\ln \left (t \right )+\frac {\pi }{3}\right ) \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 16

ode=D[y[t],{t,1}] ==(1+y[t]^2)/t; 
ic={y[1]==Sqrt[3]}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \cot \left (\frac {1}{6} (\pi -6 \log (t))\right ) \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(Derivative(y(t), t) - (y(t)**2 + 1)/t,0) 
ics = {y(1): sqrt(3)} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \tan {\left (\log {\left (t \right )} + \frac {\pi }{3} \right )} \]