68.1.25 problem 32

Internal problem ID [17092]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 32
Date solved : Thursday, October 02, 2025 at 01:43:02 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x),G(y)]`], [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 60

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

\begin{align*} y(t)&\to \frac {1}{2} \left (-\sin (t)-\sqrt {\sin ^2(t)+4 c_1}\right )\\ y(t)&\to \frac {1}{2} \left (-\sin (t)+\sqrt {\sin ^2(t)+4 c_1}\right )\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 1.149 (sec). Leaf size: 39

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

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