68.6.49 problem 55

Internal problem ID [17359]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 55
Date solved : Thursday, October 02, 2025 at 02:10:30 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} 2 t +\tan \left (y\right )+\left (t -t^{2} \tan \left (y\right )\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 134

ode:=2*t+tan(y(t))+(t-t^2*tan(y(t)))*diff(y(t),t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \arctan \left (\frac {-\sqrt {t^{4}-c_1^{2}+t^{2}}\, t -c_1}{\left (t^{2}+1\right ) t}, \frac {-c_1 t +\sqrt {t^{4}-c_1^{2}+t^{2}}}{\left (t^{2}+1\right ) t}\right ) \\ y &= \arctan \left (\frac {\sqrt {t^{4}-c_1^{2}+t^{2}}\, t -c_1}{\left (t^{2}+1\right ) t}, \frac {-c_1 t -\sqrt {t^{4}-c_1^{2}+t^{2}}}{\left (t^{2}+1\right ) t}\right ) \\ \end{align*}

✓ Mathematica. Time used: 37.003 (sec). Leaf size: 177

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

\begin{align*} y(t)&\to -\arccos \left (-\frac {c_1 t^2+\sqrt {t^6+t^4-c_1{}^2 t^2}}{t^4+t^2}\right )\\ y(t)&\to \arccos \left (-\frac {c_1 t^2+\sqrt {t^6+t^4-c_1{}^2 t^2}}{t^4+t^2}\right )\\ y(t)&\to -\arccos \left (\frac {\sqrt {t^6+t^4-c_1{}^2 t^2}-c_1 t^2}{t^4+t^2}\right )\\ y(t)&\to \arccos \left (\frac {\sqrt {t^6+t^4-c_1{}^2 t^2}-c_1 t^2}{t^4+t^2}\right ) \end{align*}

✓ Sympy. Time used: 3.667 (sec). Leaf size: 39

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

\[ \left [ y{\left (t \right )} = \operatorname {asin}{\left (\frac {C_{1}}{t \sqrt {t^{2} + 1}} \right )} - \operatorname {atan}{\left (t \right )}, \ y{\left (t \right )} = - \operatorname {asin}{\left (\frac {C_{1}}{t \sqrt {t^{2} + 1}} \right )} - \operatorname {atan}{\left (t \right )} + \pi \right ] \]