10.4.5 problem 6

Internal problem ID [1186]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.5. Page 88
Problem number : 6
Date solved : Saturday, March 29, 2025 at 10:45:07 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=-\frac {2 \arctan \left (y\right )}{1+y^{2}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 23

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

\[ t +\frac {\int _{}^{y}\frac {\textit {\_a}^{2}+1}{\arctan \left (\textit {\_a} \right )}d \textit {\_a}}{2}+c_1 = 0 \]

✓ Mathematica. Time used: 0.959 (sec). Leaf size: 38

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]^2+1}{\arctan (K[1])}dK[1]\&\right ][-2 t+c_1] \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.410 (sec). Leaf size: 15

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

\[ \int \limits ^{y{\left (t \right )}} \frac {y^{2} + 1}{\operatorname {atan}{\left (y \right )}}\, dy = C_{1} - 2 t \]