54.1.80 problem 81

Internal problem ID [11394]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 81
Date solved : Sunday, October 12, 2025 at 01:37:07 AM
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: 0.736 (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