Internal problem ID [9086]
Internal file name [OUTPUT/8021_Monday_June_06_2022_01_17_58_AM_23375380/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 752.
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.
\[ \boxed {y^{\prime }-\frac {\cos \left (y\right ) \left (\cos \left (y\right ) x^{3}-x -1\right )}{\left (x \sin \left (y\right )-1\right ) \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \cos \left (y\right )^{2} x^{3}-y^{\prime } \sin \left (y\right ) x^{2}-y^{\prime } \sin \left (y\right ) x -\cos \left (y\right ) x +y^{\prime } x -\cos \left (y\right )+y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\cos \left (y\right )^{2} x^{3}+\cos \left (y\right ) x +\cos \left (y\right )}{-\sin \left (y\right ) x^{2}-x \sin \left (y\right )+x +1} \end {array} \]
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: 0.031 (sec). Leaf size: 723
dsolve(diff(y(x),x) = cos(y(x))/(x*sin(y(x))-1)*(cos(y(x))*x^3-x-1)/(x+1),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \arctan \left (\frac {\left (2 x^{3}-3 x^{2}-6 \ln \left (x +1\right )+6 c_{1} +6 x \right ) \sqrt {36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}-36 c_{1} x^{2}+72 c_{1} x +36 c_{1}^{2}+36}+36 x}{36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}+\left (-36 c_{1} +36\right ) x^{2}+72 c_{1} x +36 c_{1}^{2}+36}, \frac {12 x^{4}-18 x^{3}-36 \ln \left (x +1\right ) x +36 c_{1} x +36 x^{2}-6 \sqrt {36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}-36 c_{1} x^{2}+72 c_{1} x +36 c_{1}^{2}+36}}{36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}+\left (-36 c_{1} +36\right ) x^{2}+72 c_{1} x +36 c_{1}^{2}+36}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\left (-2 x^{3}+3 x^{2}+6 \ln \left (x +1\right )-6 c_{1} -6 x \right ) \sqrt {36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}-36 c_{1} x^{2}+72 c_{1} x +36 c_{1}^{2}+36}+36 x}{36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}+\left (-36 c_{1} +36\right ) x^{2}+72 c_{1} x +36 c_{1}^{2}+36}, \frac {12 x^{4}-18 x^{3}-36 \ln \left (x +1\right ) x +36 c_{1} x +36 x^{2}+6 \sqrt {36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}-36 c_{1} x^{2}+72 c_{1} x +36 c_{1}^{2}+36}}{36 \ln \left (x +1\right )^{2}+\left (-24 x^{3}+36 x^{2}-72 c_{1} -72 x \right ) \ln \left (x +1\right )+4 x^{6}-12 x^{5}+33 x^{4}+\left (24 c_{1} -36\right ) x^{3}+\left (-36 c_{1} +36\right ) x^{2}+72 c_{1} x +36 c_{1}^{2}+36}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.957 (sec). Leaf size: 867
DSolve[y'[x] == (Cos[y[x]]*(-1 - x + x^3*Cos[y[x]]))/((1 + x)*(-1 + x*Sin[y[x]])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \tan ^{-1}\left (\frac {6 \left (2 x^4-3 x^3+6 x^2+\sqrt {4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 c_1 x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}-6 x \log (x+1)+6 c_1 x\right )}{4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 (-1+c_1) x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )},x-\frac {\left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ) \left (2 x^4-3 x^3+6 x^2+\sqrt {4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 c_1 x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}-6 x \log (x+1)+6 c_1 x\right )}{4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 (-1+c_1) x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}\right ) \\ y(x)\to \tan ^{-1}\left (-\frac {6 \left (-2 x^4+3 x^3-6 x^2+\sqrt {4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 c_1 x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}+6 x \log (x+1)-6 c_1 x\right )}{4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 (-1+c_1) x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )},x-\frac {\left (2 x^3-3 x^2+6 x-6 \log (x+1)+6 c_1\right ) \left (2 x^4-3 x^3+6 x^2-\sqrt {4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 c_1 x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}-6 x \log (x+1)+6 c_1 x\right )}{4 x^6-12 x^5+33 x^4+12 (-3+2 c_1) x^3-36 (-1+c_1) x^2-12 \left (2 x^3-3 x^2+6 x+6 c_1\right ) \log (x+1)+36 \log ^2(x+1)+72 c_1 x+36 \left (1+c_1{}^2\right )}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}