89.6.11 problem 11

Internal problem ID [24394]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:24:25 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 74

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

\[ y = \arctan \left (\frac {3 c_1 +3 \tan \left (x \right )}{\operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_1^{2}+18 c_1 \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )^{2}+2}, \operatorname {RootOf}\left (\textit {\_Z}^{6}+3 \textit {\_Z}^{4}+9 c_1^{2}+18 c_1 \tan \left (x \right )+9 \tan \left (x \right )^{2}-4\right )\right ) \]

✓ Mathematica. Time used: 13.945 (sec). Leaf size: 391

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

\begin{align*} y(x)&\to \arcsin \left (\frac {\sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right )\\ y(x)&\to -\arcsin \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right )\\ y(x)&\to -\arcsin \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}{2 \sqrt [3]{2}}+\frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{-3 \tan (x)+\sqrt {9 \tan ^2(x)+18 c_1 \tan (x)-4+9 c_1{}^2}-3 c_1}}\right )\\ y(x)&\to \arcsin \left (\frac {\sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}\right ) \end{align*}

✗ Sympy

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

Timed Out