60.1.81 problem 81
Internal
problem
ID
[10095]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
81
Date
solved
:
Tuesday, January 28, 2025 at 04:25:37 PM
CAS
classification
:
[`y=_G(x,y')`]
\begin{align*} y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )-1&=0 \end{align*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 78
dsolve(diff(y(x),x) + 2*tan(y(x))*tan(x) - 1=0,y(x), singsol=all)
\[
c_{1} +\frac {\tan \left (x \right )}{{\left (\frac {\left (1+\tan \left (y\right )^{2}\right ) \left (1+\tan \left (x \right )^{2}\right )}{\left (\tan \left (y\right ) \tan \left (x \right )-1\right )^{2}}\right )}^{{1}/{4}}}+\frac {\left (\tan \left (y\right )+\tan \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (\tan \left (y\right )+\tan \left (x \right )\right )^{2}}{\left (\tan \left (y\right ) \tan \left (x \right )-1\right )^{2}}\right )}{2 \tan \left (y\right ) \tan \left (x \right )-2} = 0
\]
✓ Solution by Mathematica
Time used: 1.100 (sec). Leaf size: 220
DSolve[D[y[x],x] + 2*Tan[y[x]]*Tan[x] - 1==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\left (i \cot (x)+\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}\right )^2\right )+i \tan (x)}{\sqrt [4]{-1+\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2}},y(x)\right ]
\]