60.1.80 problem 81
Internal
problem
ID
[10094]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
81
Date
solved
:
Sunday, March 30, 2025 at 03:15:34 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1&=0 \end{align*}
✓ Maple. Time used: 0.014 (sec). Leaf size: 78
ode:=diff(y(x),x)+2*tan(y(x))*tan(x)-1 = 0;
dsolve(ode,y(x), singsol=all);
\[
c_1 +\frac {\tan \left (x \right )}{{\left (\frac {\left (1+\tan \left (y\right )^{2}\right ) \left (1+\tan \left (x \right )^{2}\right )}{\left (\tan \left (y\right ) \tan \left (x \right )-1\right )^{2}}\right )}^{{1}/{4}}}+\frac {\left (\tan \left (y\right )+\tan \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (\tan \left (y\right )+\tan \left (x \right )\right )^{2}}{\left (\tan \left (y\right ) \tan \left (x \right )-1\right )^{2}}\right )}{2 \tan \left (y\right ) \tan \left (x \right )-2} = 0
\]
✓ Mathematica. Time used: 1.1 (sec). Leaf size: 220
ode=D[y[x],x] + 2*Tan[y[x]]*Tan[x] - 1==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\left (i \cot (x)+\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}\right )^2\right )+i \tan (x)}{\sqrt [4]{-1+\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2}},y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*tan(x)*tan(y(x)) + Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE 2*tan(x)*tan(y(x)) + Derivative(y(x), x) - 1 cannot be solved by the lie group method