Internal problem ID [8430]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 850.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {1}{\sin \relax (x )}-f_{1}\left (y-\ln \left (\sin \relax (x )\right )+\ln \left (\cos \relax (x )+1\right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.443 (sec). Leaf size: 32
dsolve(diff(y(x),x) = 1/sin(x)+_F1(y(x)-ln(sin(x))+ln(cos(x)+1)),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \relax (x )}\frac {1}{f_{1}\left (\textit {\_a} -\ln \left (\sin \relax (x )\right )+\ln \left (1+\cos \relax (x )\right )\right )}d \textit {\_a} -x -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.47 (sec). Leaf size: 1438
DSolve[y'[x] == Csc[x] + F1[Log[1 + Cos[x]] - Log[Sin[x]] + y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x))) \sin (K[1])}{-\cot ^2(K[1])+\text {F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x)) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(\log (\cos (K[1])+1)-\log (\sin (K[1]))+y(x))-1}dK[1]+\int _1^{y(x)}-\frac {\sin (x) \left (\int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \sin (K[1]) \left (\cot (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right )^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc ^3(x)+\text {F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \sin (K[1]) \left (\cot (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right )^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc ^2(x)-\cot (x) \csc (x)-\cot ^2(x) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \sin (K[1]) \left (\cot (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right )^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)+\cot (x) \text {F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \sin (K[1]) \left (\cot (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right )^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)-\int _1^x\left (\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) (\csc (K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))) \sin (K[1]) \left (\cot (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))+\csc (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))\right )}{\left (-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1\right )^2}-\frac {\left (\cot ^2(K[1])+\csc (K[1]) \cot (K[1])+1\right ) \sin (K[1]) \text {F1}'(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))}{-\cot ^2(K[1])+\text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1]))) \cot (K[1])+\csc ^2(K[1])+\csc (K[1]) \text {F1}(K[2]+\log (\cos (K[1])+1)-\log (\sin (K[1])))-1}\right )dK[1] \csc (x)-\cot ^2(x)-1\right )}{-\cot ^2(x)+\text {F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x))) \cot (x)+\csc ^2(x)+\csc (x) \text {F1}(K[2]+\log (\cos (x)+1)-\log (\sin (x)))-1}dK[2]=c_1,y(x)\right ] \]