23.1.424 problem 414

Internal problem ID [5031]
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 : 414
Date solved : Tuesday, September 30, 2025 at 11:27:53 AM
CAS classification : [_separable]

\begin{align*} \left (1-4 \cos \left (x \right )^{2}\right ) y^{\prime }&=\tan \left (x \right ) \left (1+4 \cos \left (x \right )^{2}\right ) y \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 15

ode:=(1-4*cos(x)^2)*diff(y(x),x) = tan(x)*(1+4*cos(x)^2)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (4 \cos \left (x \right )-\sec \left (x \right )\right ) \]

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 48

ode=(1-4*Cos[x]^2)*D[y[x],x]==Tan[x](1+4*Cos[x]^2)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \exp \left (\int _1^x-\frac {2 \sin (K[1])+\sin (3 K[1])}{2 \cos (K[1])+\cos (3 K[1])}dK[1]\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 4.273 (sec). Leaf size: 112

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 4*cos(x)**2)*Derivative(y(x), x) - (4*cos(x)**2 + 1)*y(x)*tan(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} \left (\tan ^{8}{\left (\frac {x}{2} \right )} - 4 \tan ^{6}{\left (\frac {x}{2} \right )} + 6 \tan ^{4}{\left (\frac {x}{2} \right )} - 4 \tan ^{2}{\left (\frac {x}{2} \right )} + 1\right ) e^{- 5 \int \frac {\tan {\left (x \right )}}{\left (2 \cos {\left (x \right )} - 1\right ) \left (2 \cos {\left (x \right )} + 1\right )}\, dx}}{\sqrt {\tan ^{2}{\left (\frac {x}{2} \right )} - 3} \sqrt {3 \tan ^{2}{\left (\frac {x}{2} \right )} - 1} \left (3 \tan ^{6}{\left (\frac {x}{2} \right )} - 7 \tan ^{4}{\left (\frac {x}{2} \right )} - 7 \tan ^{2}{\left (\frac {x}{2} \right )} + 3\right )} \]