76.1.3 problem 3

Internal problem ID [17231]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 3
Date solved : Monday, March 31, 2025 at 03:45:14 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+y^{3} \sin \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

ode:=diff(y(x),x)+y(x)^3*sin(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1}{\sqrt {c_1 -2 \cos \left (x \right )}} \\ y &= -\frac {1}{\sqrt {c_1 -2 \cos \left (x \right )}} \\ \end{align*}

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 73

ode=D[y[x],x]+y[x]^3*Sin[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {1}{\sqrt {2} \sqrt {-\int _1^x-\sin (K[1])dK[1]-c_1}} \\ y(x)\to \frac {1}{\sqrt {2} \sqrt {-\int _1^x-\sin (K[1])dK[1]-c_1}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.480 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**3*sin(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} + \cos {\left (x \right )}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \frac {1}{C_{1} + \cos {\left (x \right )}}}}{2}\right ] \]