57.4.10 problem 4(b)

Internal problem ID [14334]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number : 4(b)
Date solved : Thursday, October 02, 2025 at 09:31:42 AM
CAS classification : [_separable]

\begin{align*} \theta ^{\prime }&=t \sqrt {t^{2}+1}\, \sec \left (\theta \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

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

\[ \theta = \arcsin \left (\frac {t^{2} \sqrt {t^{2}+1}}{3}+\frac {\sqrt {t^{2}+1}}{3}+c_1 \right ) \]

✓ Mathematica. Time used: 3.879 (sec). Leaf size: 73

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

\begin{align*} \theta (t)&\to \arcsin \left (\frac {1}{3} \left (\sqrt {t^2+1} t^2+\sqrt {t^2+1}+3 c_1\right )\right )\\ \theta (t)&\to \arcsin \left (\frac {1}{3} \left (\sqrt {t^2+1} t^2+\sqrt {t^2+1}+3 c_1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.264 (sec). Leaf size: 60

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

\[ \left [ \theta {\left (t \right )} = \pi - \operatorname {asin}{\left (C_{1} + \frac {t^{2} \sqrt {t^{2} + 1}}{3} + \frac {\sqrt {t^{2} + 1}}{3} \right )}, \ \theta {\left (t \right )} = \operatorname {asin}{\left (C_{1} + \frac {t^{2} \sqrt {t^{2} + 1}}{3} + \frac {\sqrt {t^{2} + 1}}{3} \right )}\right ] \]