Internal problem ID [7445]
Internal file name [OUTPUT/6412_Sunday_June_05_2022_04_47_26_PM_44867493/index.tex]
Book: Second order enumerated odes
Section: section 2
Problem number: 4.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+\left (\sin \left (x \right )+2 x \right ) y^{\prime }+\cos \left (y\right ) y {y^{\prime }}^{2}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 34
dsolve(diff(y(x),x$2)+(sin(x)+2*x)*diff(y(x),x)+cos(y(x))*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
\[ \int _{}^{y \left (x \right )}{\mathrm e}^{\cos \left (\textit {\_a} \right )+\sin \left (\textit {\_a} \right ) \textit {\_a}}d \textit {\_a} -c_{1} \left (\int {\mathrm e}^{-x^{2}+\cos \left (x \right )}d x \right )-c_{2} = 0 \]
✓ Solution by Mathematica
Time used: 1.16 (sec). Leaf size: 53
DSolve[y''[x]+(Sin[x]+2*x)*y'[x]+Cos[y[x]]*y[x]*(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}e^{\cos (K[1])+K[1] \sin (K[1])}dK[1]\&\right ]\left [\int _1^x-e^{\cos (K[2])-K[2]^2} c_1dK[2]+c_2\right ] \]