78.4.5 problem 6

Internal problem ID [18128]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Section 8 (Exact Equations). Problems at page 72
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 11:30:08 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 31

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

\begin{align*} y &= \frac {\pi }{2} \\ y &= \arccos \left (\sqrt {c_1 \sin \left (x \right )}\right ) \\ y &= \frac {\pi }{2}+\arcsin \left (\sqrt {c_1 \sin \left (x \right )}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 5.061 (sec). Leaf size: 73

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

\begin{align*} y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ y(x)\to -\arccos \left (-\frac {1}{4} c_1 \sqrt {\sin (x)}\right ) \\ y(x)\to \arccos \left (-\frac {1}{4} c_1 \sqrt {\sin (x)}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}