89.5.21 problem 21

Internal problem ID [24371]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 43
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:22:25 PM
CAS classification : [_linear]

\begin{align*} \left (1+\cos \left (x \right )\right ) y^{\prime }&=\sin \left (x \right ) \left (\sin \left (x \right )+\sin \left (x \right ) \cos \left (x \right )-y\right ) \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

ode:=(cos(x)+1)*diff(y(x),x) = sin(x)*(sin(x)+cos(x)*sin(x)-y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x -\sin \left (x \right )+c_1 \right ) \left (1+\cos \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.062 (sec). Leaf size: 24

ode=(1+Cos[x])*D[y[x],x]== Sin[x]*(Sin[x]+Sin[x]*Cos[x]-y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos ^2\left (\frac {x}{2}\right ) (2 x-2 \sin (x)+c_1) \end{align*}

✓ Sympy. Time used: 91.222 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((cos(x) + 1)*Derivative(y(x), x) - (-y(x) + sin(x)*cos(x) + sin(x))*sin(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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