68.6.18 problem 19

Internal problem ID [17328]
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 : 19
Date solved : Thursday, October 02, 2025 at 02:02:43 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.120 (sec). Leaf size: 21

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

\begin{align*} y &= \arcsin \left (\frac {1}{\sqrt {c_1 t}}\right ) \\ y &= -\arcsin \left (\frac {1}{\sqrt {c_1 t}}\right ) \\ \end{align*}

✓ Mathematica. Time used: 4.1 (sec). Leaf size: 26

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

\begin{align*} y(t)&\to 0\\ y(t)&\to \arcsin \left (\frac {e^{c_1}}{\sqrt {t}}\right )\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 0.232 (sec). Leaf size: 22

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

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