89.10.7 problem 7

Internal problem ID [24500]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 77
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:43:10 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} y^{3} \sec \left (x \right )^{2}-\left (1-2 y^{2} \tan \left (x \right )\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 53

ode:=y(x)^3*sec(x)^2-(1-2*y(x)^2*tan(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ -\frac {y^{2}}{-1+2 y^{2} \tan \left (x \right )} = -\frac {\left ({\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}} \ln \left (-\frac {i \left ({\mathrm e}^{\textit {\_Z}}+9\right ) \cot \left (x \right )}{2}\right )+3 c_1 \,{\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+9\right )}+9\right ) \cot \left (x \right )}{18} \]

✓ Mathematica. Time used: 3.405 (sec). Leaf size: 74

ode=( y[x]^3*Sec[x]^2)-( 1-2*y[x]^2*Tan[x]   )*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}}\\ y(x)&\to \frac {i \sqrt {\cot (x)} \sqrt {W\left (-2 e^{-8 c_1} \tan (x)\right )}}{\sqrt {2}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(-2*y(x)**2*tan(x) + 1)*Derivative(y(x), x) + y(x)**3*sec(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out