82.12.4 problem Ex. 4

Internal problem ID [18769]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 4
Date solved : Tuesday, January 28, 2025 at 12:15:56 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: 0.074 (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: 0.419 (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*}