Internal problem ID [7662]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 81.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }+2 \tan \relax (y) \tan \relax (x )-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 78
dsolve(diff(y(x),x) + 2*tan(y(x))*tan(x) - 1=0,y(x), singsol=all)
\[ c_{1}+\frac {\tan \relax (x )}{\left (\frac {\left (1+\tan ^{2}\left (y \relax (x )\right )\right ) \left (1+\tan ^{2}\relax (x )\right )}{\left (\tan \left (y \relax (x )\right ) \tan \relax (x )-1\right )^{2}}\right )^{\frac {1}{4}}}+\frac {\left (\tan \left (y \relax (x )\right )+\tan \relax (x )\right ) \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (\tan \left (y \relax (x )\right )+\tan \relax (x )\right )^{2}}{\left (\tan \left (y \relax (x )\right ) \tan \relax (x )-1\right )^{2}}\right )}{2 \tan \left (y \relax (x )\right ) \tan \relax (x )-2} = 0 \]
✓ Solution by Mathematica
Time used: 1.832 (sec). Leaf size: 220
DSolve[y'[x] + 2*Tan[y[x]]*Tan[x] - 1==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {3}{2};\left (i \cot (x)+\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}\right )^2\right )+i \tan (x)}{\sqrt [4]{-1+\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2}},y(x)\right ] \]