Internal problem ID [9076]
Internal file name [OUTPUT/8011_Monday_June_06_2022_01_15_29_AM_70508125/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 742.
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.
\[ \boxed {y^{\prime }+\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (x \sin \left (y\right )-1\right ) \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sin \left (y\right ) x^{2}+y^{\prime } \sin \left (y\right ) x -y^{\prime } x -\cos \left (y\right )^{2}+x \cos \left (y\right )-y^{\prime }+\cos \left (y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cos \left (y\right )^{2}-x \cos \left (y\right )-\cos \left (y\right )}{\sin \left (y\right ) x^{2}+x \sin \left (y\right )-x -1} \end {array} \]
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`[0, (1+cos(2*y))/(x*sin(y)-1)]
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 239
dsolve(diff(y(x),x) = -cos(y(x))/(x*sin(y(x))-1)*(x-cos(y(x))+1)/(x+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \arctan \left (\frac {\left (-\ln \left (x +1\right )+c_{1} \right ) \sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}+x}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}, \frac {\ln \left (x +1\right ) x -c_{1} x +\sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\left (\ln \left (x +1\right )-c_{1} \right ) \sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}+x}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}, \frac {\ln \left (x +1\right ) x -c_{1} x -\sqrt {\ln \left (x +1\right )^{2}-2 c_{1} \ln \left (x +1\right )+c_{1}^{2}-x^{2}+1}}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 51.98 (sec). Leaf size: 315
DSolve[y'[x] == -(((1 + x - Cos[y[x]])*Cos[y[x]])/((1 + x)*(-1 + x*Sin[y[x]]))),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sec ^{-1}\left (\frac {-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\ y(x)\to \sec ^{-1}\left (\frac {-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\ y(x)\to -\sec ^{-1}\left (\frac {\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\ y(x)\to \sec ^{-1}\left (\frac {\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}+x \log (x+1)+c_1 x}{x^2-1}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ y(x)\to \sec ^{-1}\left (\frac {x \log (x+1)-\sqrt {-x^2+\log ^2(x+1)+1}}{x^2-1}\right ) \\ y(x)\to \sec ^{-1}\left (\frac {\sqrt {-x^2+\log ^2(x+1)+1}+x \log (x+1)}{x^2-1}\right ) \\ \end{align*}