Internal problem ID [8686]
Internal file name [OUTPUT/7619_Sunday_June_05_2022_11_20_22_PM_71209624/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 350.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
unknown
Unable to solve or complete the solution.
\[ \boxed {y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right )^{2}-\sin \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cos \left (x \right ) \sin \left (y\right )^{2}+\sin \left (y\right )}{\cos \left (y\right )} \end {array} \]
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)] <- symmetry pattern of the form [0, F(x)*G(y)] successful`
✓ Solution by Maple
Time used: 0.437 (sec). Leaf size: 266
dsolve(diff(y(x),x)*cos(y(x))-cos(x)*sin(y(x))^2-sin(y(x)) = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \arctan \left (-\frac {2 \,{\mathrm e}^{x}}{\left (\sin \left (x \right )+\cos \left (x \right )\right ) {\mathrm e}^{x}+2 c_{1}}, \frac {\sqrt {\left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1}^{2}+{\mathrm e}^{2 x}\right ) \left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_{1}^{2}\right )}}{2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1}^{2}+{\mathrm e}^{2 x}}\right ) \\ y \left (x \right ) &= \arctan \left (-\frac {2 \,{\mathrm e}^{x}}{\left (\sin \left (x \right )+\cos \left (x \right )\right ) {\mathrm e}^{x}+2 c_{1}}, -\frac {\sqrt {\left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1}^{2}+{\mathrm e}^{2 x}\right ) \left (2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}-3 \,{\mathrm e}^{2 x}+4 c_{1}^{2}\right )}}{2 \cos \left (x \right ) \sin \left (x \right ) {\mathrm e}^{2 x}+4 c_{1} \sin \left (x \right ) {\mathrm e}^{x}+4 \cos \left (x \right ) c_{1} {\mathrm e}^{x}+4 c_{1}^{2}+{\mathrm e}^{2 x}}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.958 (sec). Leaf size: 58
DSolve[-Sin[y[x]] - Cos[x]*Sin[y[x]]^2 + Cos[y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \csc ^{-1}\left (\frac {1}{2} \left (-\sin (x)-\cos (x)-2 c_1 e^{-x}\right )\right ) \\ y(x)\to -\csc ^{-1}\left (\frac {1}{2} \left (\sin (x)+\cos (x)+2 c_1 e^{-x}\right )\right ) \\ y(x)\to 0 \\ \end{align*}