60.2.274 problem 850

Internal problem ID [10861]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 850
Date solved : Tuesday, January 28, 2025 at 05:26:32 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y^{\prime }&=\frac {1}{\sin \left (x \right )}+\textit {\_F1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right ) \end{align*}

✓ Solution by Maple

Time used: 0.233 (sec). Leaf size: 27

dsolve(diff(y(x),x) = 1/sin(x)+_F1(y(x)-ln(sin(x))+ln(cos(x)+1)),y(x), singsol=all)
 

\[ y = -\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.545 (sec). Leaf size: 1438

DSolve[D[y[x],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 ] \]