29.5.7 problem 122
Internal
problem
ID
[4724]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
5
Problem
number
:
122
Date
solved
:
Sunday, March 30, 2025 at 03:49:36 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\sec \left (x \right )^{2} \sec \left (y\right )^{3} \end{align*}
✓ Maple. Time used: 0.023 (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: 21.167 (sec). Leaf size: 565
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 ) \\
y(x)\to -\arcsin \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}\right ) \\
y(x)\to -\arcsin \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {9 \tan ^2(x)-4}-3 \tan (x)}}{2 \sqrt [3]{2}}+\frac {1-i \sqrt {3}}{2^{2/3} \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(Derivative(y(x), x) - 1/(cos(x)**2*cos(y(x))**3),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
ZeroDivisionError : polynomial division