15.8.21 problem 22

Internal problem ID [3024]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 22
Date solved : Monday, January 27, 2025 at 07:09:13 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.210 (sec). Leaf size: 69

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

\[ y = \arctan \left (\frac {c_{1}^{2} {\mathrm e}^{x +\frac {\sin \left (2 x \right )}{2}}-1}{c_{1}^{2} {\mathrm e}^{x +\frac {\sin \left (2 x \right )}{2}}+1}, \frac {2 c_{1} {\mathrm e}^{\frac {x}{2}+\frac {\sin \left (2 x \right )}{4}}}{c_{1}^{2} {\mathrm e}^{x +\frac {\sin \left (2 x \right )}{2}}+1}\right ) \]

✓ Solution by Mathematica

Time used: 0.995 (sec). Leaf size: 41

DSolve[D[y[x],x]==Cos[y[x]]*Cos[x]^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 2 \arctan \left (\tanh \left (\frac {1}{8} (2 x+\sin (2 x)+c_1)\right )\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}