89.7.2 problem 2

Internal problem ID [24433]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 2
Date solved : Thursday, October 02, 2025 at 10:29:37 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 17

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

\[ y = \arcsin \left (\frac {3 x}{-1+c_1 \,x^{{3}/{2}}}\right ) \]

✓ Mathematica. Time used: 0.388 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \csc ^{-1}\left (\frac {-4-3 c_1 x^{3/2}}{12 x}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.971 (sec). Leaf size: 42

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {3 x^{\frac {3}{2}}}{C_{1} x^{2} - \sqrt {x}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {3 x^{\frac {3}{2}}}{C_{1} x^{2} - \sqrt {x}} \right )}\right ] \]