31.1.6 problem 1.6

Internal problem ID [5704]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 1.6
Date solved : Monday, January 27, 2025 at 01:08:27 PM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.270 (sec). Leaf size: 47

dsolve(sec(x)^2*tan(y(x))+sec(y(x))^2*tan(x)*diff(y(x),x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.477 (sec). Leaf size: 68

DSolve[Sec[x]^2*Tan[y[x]]+Sec[y[x]]^2*Tan[x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1)) \\ y(x)\to \frac {1}{2} \arccos (-\tanh (\text {arctanh}(\cos (2 x))+2 c_1)) \\ y(x)\to 0 \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}