Internal problem ID [9077]
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')`]
\[ \boxed {y^{\prime }+\frac {\cos \left (y\right ) \left (x -\cos \left (y\right )+1\right )}{\left (\sin \left (y\right ) x -1\right ) \left (1+x \right )}=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 373
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 {\ln \left (x +1\right ) \left (\ln \left (x +1\right ) x -x c_{1} -\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 -x c_{1} -\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 -x c_{1} -\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 {\ln \left (x +1\right ) \left (\ln \left (x +1\right ) x -x c_{1} +\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 -x c_{1} +\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 -x c_{1} +\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*}