76.4.18 problem 22

Internal problem ID [17341]
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 2.6 (Exact equations and integrating factors). Problems at page 100
Problem number : 22
Date solved : Monday, March 31, 2025 at 03:55:17 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 16

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

\[ y = \arcsin \left (\frac {{\mathrm e}^{-x}}{c_1 \,x^{2}}\right ) \]

✓ Mathematica. Time used: 46.669 (sec). Leaf size: 24

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

\begin{align*} y(x)\to \arcsin \left (\frac {e^{-x-2+c_1}}{x^2}\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.444 (sec). Leaf size: 26

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {C_{1} e^{- x}}{x^{2}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {C_{1} e^{- x}}{x^{2}} \right )}\right ] \]