89.1.15 problem 15

Internal problem ID [24250]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 15
Date solved : Thursday, October 02, 2025 at 10:01:58 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 23

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

\[ \theta = \frac {\operatorname {RootOf}\left (\textit {\_Z} +z^{4}-2 z^{2}+4 c_1 +\sin \left (\textit {\_Z} \right )+1\right )}{2} \]

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 38

ode=D[\[Theta][z],z]==z*(1-z^2)*Sec[ \[Theta][z] ]^2; 
ic={}; 
DSolve[{ode,ic},\[Theta][z],z,IncludeSingularSolutions->True]
 

\begin{align*} \theta (z)&\to \text {InverseFunction}\left [2 \left (\frac {\text {$\#$1}}{2}+\frac {1}{4} \sin (2 \text {$\#$1})\right )\&\right ]\left [-\frac {z^4}{2}+z^2+c_1\right ] \end{align*}

✓ Sympy. Time used: 1.403 (sec). Leaf size: 27

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

\[ \frac {z^{4}}{4} - \frac {z^{2}}{2} + \frac {y{\left (z \right )}}{2} + \frac {\sin {\left (y{\left (z \right )} \right )} \cos {\left (y{\left (z \right )} \right )}}{2} = C_{1} \]