Internal problem ID [8322]
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.
CAS Maple gives this as type [y=_G(x,y')]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {\cos \relax (y) \left (x -\cos \relax (y)+1\right )}{\left (x \sin \relax (y)-1\right ) \left (x +1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.48 (sec). Leaf size: 368
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 \relax (x ) = \arctan \left (-\frac {\ln \left (x +1\right ) \left (\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}\right )}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}+\frac {c_{1} \left (\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}\right )}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}+x , \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 \relax (x ) = \arctan \left (\frac {\ln \left (x +1\right ) \left (-\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}\right )}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}-\frac {c_{1} \left (-\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}\right )}{c_{1}^{2}-2 c_{1} \ln \left (x +1\right )+\ln \left (x +1\right )^{2}+1}+x , -\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: 82.021 (sec). Leaf size: 313
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 {x (\log (x+1)+c_1)-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}}{x^2-1}\right ) \\ y(x)\to \sec ^{-1}\left (\frac {x (\log (x+1)+c_1)-\sqrt {-x^2+\log ^2(x+1)+2 c_1 \log (x+1)+1+c_1{}^2}}{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^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^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*}