64.3.6 problem 7

Internal problem ID [13277]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 05:14:01 AM
CAS classification : [_exact, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.111 (sec). Leaf size: 61

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

\begin{align*} y &= -\frac {\tan \left (x \right )}{2}-\frac {\sec \left (x \right ) \sqrt {-4 c_{1} \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-4 \cos \left (x \right )}}{2} \\ y &= -\frac {\tan \left (x \right )}{2}+\frac {\sec \left (x \right ) \sqrt {-4 c_{1} \cos \left (x \right )^{2}+\sin \left (x \right )^{2}-4 \cos \left (x \right )}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.034 (sec). Leaf size: 101

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

\begin{align*} y(x)\to \frac {1}{4} \left (-2 \tan (x)-\sqrt {2} \sqrt {\sec ^2(x)} \sqrt {-8 \cos (x)+(-1+4 c_1) \cos (2 x)+1+4 c_1}\right ) \\ y(x)\to \frac {1}{4} \left (-2 \tan (x)+\sqrt {\sec ^2(x)} \sqrt {-16 \cos (x)+(-2+8 c_1) \cos (2 x)+2+8 c_1}\right ) \\ \end{align*}