Internal problem ID [9140]
Internal file name [OUTPUT/8075_Monday_June_06_2022_01_40_19_AM_41924801/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 806.
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.
\[ \boxed {y^{\prime }-\frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 y^{\prime } x^{2}+2 y^{\prime } x +\sin \left (2 y\right ) x -x \cos \left (2 y\right )+\sin \left (2 y\right )-x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\sin \left (2 y\right ) x -\sin \left (2 y\right )+x \cos \left (2 y\right )+x}{2 x^{2}+2 x} \end {array} \]
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]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve(diff(y(x),x) = 1/2*(-sin(2*y(x))*x-sin(2*y(x))+x*cos(2*y(x))+x)/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = -\arctan \left (\frac {\ln \left (x +1\right )-x -c_{1}}{x}\right ) \]
✓ Solution by Mathematica
Time used: 2.411 (sec). Leaf size: 56
DSolve[y'[x] == (x/2 + (x*Cos[2*y[x]])/2 - Sin[2*y[x]]/2 - (x*Sin[2*y[x]])/2)/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \arctan \left (\frac {x-\log (x+1)-c_1}{x}\right ) \\ y(x)\to -\frac {1}{2} \pi \sqrt {\frac {1}{x^2}} x \\ y(x)\to \frac {1}{2} \pi \sqrt {\frac {1}{x^2}} x \\ \end{align*}