68.4.18 problem 18

Internal problem ID [17198]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.2, page 39
Problem number : 18
Date solved : Thursday, October 02, 2025 at 01:51:03 PM
CAS classification : [_separable]

\begin{align*} \sin \left (t \right )^{2}&=\cos \left (y\right )^{2} y^{\prime } \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 27

ode:=sin(t)^2 = cos(y(t))^2*diff(y(t),t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\operatorname {RootOf}\left (-\textit {\_Z} +2 t +4 c_1 -\sin \left (2 t \right )-\sin \left (\textit {\_Z} \right )\right )}{2} \]

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 39

ode=Sin[t]^2==Cos[y[t]]^2*D[y[t],t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}(\cos (2 K[1])+1)dK[1]\&\right ]\left [\int _1^t2 \sin ^2(K[2])dK[2]+c_1\right ] \end{align*}

✓ Sympy. Time used: 20.970 (sec). Leaf size: 29

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

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