Internal problem ID [8418]
Internal file name [OUTPUT/7351_Sunday_June_05_2022_10_53_15_PM_90987742/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+2 \tan \left (y\right ) \tan \left (x \right )=1 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-2 \tan \left (y\right ) \tan \left (x \right )+1 \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] `, `-> Computing symmetries using: way = HINT -> Calling odsolve with the ODE`, diff(y(x), x) = y(x)*(1+tan(x)^2)/tan(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, -2*K[1]*tan(x)+diff(y(x), x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful -> Calling odsolve with the ODE`, diff(y(x), x)+y(x)*(1+tan(x)^2)/tan(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)-K[1], y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+(tan(x)^2*y(x)-K[1]*tan(x)+y(x))/tan(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x) = -y(x)*(1+tan(x)^2)/tan(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, 2*K[1]*tan(x)+diff(y(x), x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful -> Calling odsolve with the ODE`, diff(y(x), x)-y(x)*(1+tan(x)^2)/tan(x), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> Calling odsolve with the ODE`, diff(y(x), x)+K[1], y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✓ 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)
\[ c_{1} +\frac {\tan \left (x \right )}{{\left (\frac {\left (1+\tan \left (y \left (x \right )\right )^{2}\right ) \left (1+\tan \left (x \right )^{2}\right )}{\left (\tan \left (y \left (x \right )\right ) \tan \left (x \right )-1\right )^{2}}\right )}^{\frac {1}{4}}}+\frac {\left (\tan \left (y \left (x \right )\right )+\tan \left (x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (\tan \left (y \left (x \right )\right )+\tan \left (x \right )\right )^{2}}{\left (\tan \left (y \left (x \right )\right ) \tan \left (x \right )-1\right )^{2}}\right )}{2 \tan \left (y \left (x \right )\right ) \tan \left (x \right )-2} = 0 \]
✓ Solution by Mathematica
Time used: 1.262 (sec). Leaf size: 220
DSolve[y'[x] + 2*Tan[y[x]]*Tan[x] - 1==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=\frac {\frac {1}{2} \left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right ) \sqrt [4]{1-\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5}{4},\frac {3}{2},\left (i \cot (x)+\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}\right )^2\right )+i \tan (x)}{\sqrt [4]{-1+\left (\frac {1}{\frac {i \tan (x)}{\tan ^2(x)+1}-\frac {i \tan ^2(x) \tan (y(x))}{\tan ^2(x)+1}}+i \cot (x)\right )^2}},y(x)\right ] \]