36.1.12 problem 12

Internal problem ID [6267]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page 46
Problem number : 12
Date solved : Monday, January 27, 2025 at 01:50:39 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {\sec \left (y\right )^{2}}{x^{2}+1} \end{align*}

✓ Solution by Maple

Time used: 0.094 (sec). Leaf size: 81

dsolve(diff(y(x),x)=sec(y(x))^2/(1+x^2),y(x), singsol=all)
 

\[ y = \frac {\arcsin \left (\operatorname {RootOf}\left (\textit {\_Z} +2 x^{2} \textit {\_Z} +\textit {\_Z} \,x^{4}-x^{4} \sin \left (-\textit {\_Z} +4 c_{1} \right )+4 x^{3} \cos \left (-\textit {\_Z} +4 c_{1} \right )+6 x^{2} \sin \left (-\textit {\_Z} +4 c_{1} \right )-4 x \cos \left (-\textit {\_Z} +4 c_{1} \right )-\sin \left (-\textit {\_Z} +4 c_{1} \right )\right )\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.584 (sec). Leaf size: 32

DSolve[D[y[x],x]==Sec[y[x]]^2/(1+x^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [2 \left (\frac {\text {$\#$1}}{2}+\frac {1}{4} \sin (2 \text {$\#$1})\right )\&\right ][2 \arctan (x)+c_1] \]