19.1.15 problem 15

Internal problem ID [3529]
Book : Differential equations and linear algebra, Stephen W. Goode, second edition, 2000
Section : 1.4, page 36
Problem number : 15
Date solved : Sunday, March 30, 2025 at 01:46:12 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 25

ode:=diff(y(x),x) = y(x)^3*sin(x); 
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: 0.17 (sec). Leaf size: 49

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

\begin{align*} y(x)\to -\frac {1}{\sqrt {2} \sqrt {\cos (x)-c_1}} \\ y(x)\to \frac {1}{\sqrt {2} \sqrt {\cos (x)-c_1}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.499 (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)
 

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