15.7.15 problem 15

Internal problem ID [2996]
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 : 15
Date solved : Sunday, March 30, 2025 at 01:04:14 AM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right ) y^{\prime }+\left (\sin \left (y\right )-1\right ) \cos \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 16

ode:=cos(y(x))*diff(y(x),x)+(sin(y(x))-1)*cos(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 60.341 (sec). Leaf size: 225

ode=Cos[y[x]]*D[y[x],x]+(Sin[y[x]]-1)*Cos[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\frac {3 \pi }{2} \\ y(x)\to \frac {\pi }{2} \\ y(x)\to -2 \arccos \left (-\frac {1}{8} e^{-\sin (x)} \left (c_1 e^{\frac {\sin (x)}{2}}+\sqrt {e^{\sin (x)} \left (32 e^{\sin (x)}-c_1{}^2\right )}\right )\right ) \\ y(x)\to 2 \arccos \left (-\frac {1}{8} e^{-\sin (x)} \left (c_1 e^{\frac {\sin (x)}{2}}+\sqrt {e^{\sin (x)} \left (32 e^{\sin (x)}-c_1{}^2\right )}\right )\right ) \\ y(x)\to -2 \arccos \left (\frac {1}{8} e^{-\sin (x)} \left (\sqrt {e^{\sin (x)} \left (32 e^{\sin (x)}-c_1{}^2\right )}-c_1 e^{\frac {\sin (x)}{2}}\right )\right ) \\ y(x)\to 2 \arccos \left (\frac {1}{8} e^{-\sin (x)} \left (\sqrt {e^{\sin (x)} \left (32 e^{\sin (x)}-c_1{}^2\right )}-c_1 e^{\frac {\sin (x)}{2}}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.728 (sec). Leaf size: 19

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

\[ \int \limits ^{y{\left (x \right )}} \frac {\cos {\left (y \right )}}{\cos {\left (y \right )} \tan {\left (y \right )} - 1}\, dy = C_{1} - \sin {\left (x \right )} \]