15.7.17 problem 17

Internal problem ID [2998]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 17
Date solved : Sunday, March 30, 2025 at 01:05:01 AM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 86

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

\begin{align*} y &= -\frac {\sqrt {\left (c_1 -2 \int {\mathrm e}^{-2 \sin \left (x \right )} \sin \left (x \right )d x \right ) {\mathrm e}^{-2 \sin \left (x \right )}}}{c_1 -2 \int {\mathrm e}^{-2 \sin \left (x \right )} \sin \left (x \right )d x} \\ y &= \frac {\sqrt {\left (c_1 -2 \int {\mathrm e}^{-2 \sin \left (x \right )} \sin \left (x \right )d x \right ) {\mathrm e}^{-2 \sin \left (x \right )}}}{c_1 -2 \int {\mathrm e}^{-2 \sin \left (x \right )} \sin \left (x \right )d x} \\ \end{align*}

✓ Mathematica. Time used: 10.405 (sec). Leaf size: 84

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

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

✓ Sympy. Time used: 88.422 (sec). Leaf size: 60

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

\[ \left [ y{\left (x \right )} = - \sqrt {\frac {e^{- 2 \sin {\left (x \right )}}}{C_{1} - 2 \int e^{- 2 \sin {\left (x \right )}} \sin {\left (x \right )}\, dx}}, \ y{\left (x \right )} = \sqrt {\frac {e^{- 2 \sin {\left (x \right )}}}{C_{1} - 2 \int e^{- 2 \sin {\left (x \right )}} \sin {\left (x \right )}\, dx}}\right ] \]