23.1.95 problem 92

Internal problem ID [4702]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 92
Date solved : Tuesday, September 30, 2025 at 08:15:34 AM
CAS classification : [_separable]

\begin{align*} y^{\prime }+y^{3} \sec \left (x \right ) \tan \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 48

ode:=diff(y(x),x)+y(x)^3*sec(x)*tan(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {\left (c_1 \cos \left (x \right )+2\right ) \cos \left (x \right )}}{c_1 \cos \left (x \right )+2} \\ y &= -\frac {\sqrt {\left (c_1 \cos \left (x \right )+2\right ) \cos \left (x \right )}}{c_1 \cos \left (x \right )+2} \\ \end{align*}

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 49

ode=D[y[x],x]+y[x]^3*Sec[x]*Tan[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{\sqrt {2} \sqrt {\sec (x)-c_1}}\\ y(x)&\to \frac {1}{\sqrt {2} \sqrt {\sec (x)-c_1}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.419 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)**3*tan(x)/cos(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- \frac {\cos {\left (x \right )}}{C_{1} \cos {\left (x \right )} - 1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- \frac {\cos {\left (x \right )}}{C_{1} \cos {\left (x \right )} - 1}}}{2}\right ] \]