65.3.2 problem 8.1 (ii)

Internal problem ID [13641]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 8, Separable equations. Exercises page 72
Problem number : 8.1 (ii)
Date solved : Wednesday, March 05, 2025 at 10:06:07 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\left (1+y^{2}\right ) \tan \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 12

ode:=diff(y(x),x) = (1+y(x)^2)*tan(x); 
ic:=y(0) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \cot \left (\frac {\pi }{4}+\ln \left (\cos \left (x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.249 (sec). Leaf size: 15

ode=D[y[x],x]==(1+y[x]^2)*Tan[x]; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \cot \left (\log (\cos (x))+\frac {\pi }{4}\right ) \]

✓ Sympy. Time used: 0.314 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \cot {\left (\log {\left (\cos {\left (x \right )} \right )} + \frac {\pi }{4} \right )} \]