Internal problem ID [9184]
Internal file name [OUTPUT/8119_Monday_June_06_2022_01_49_07_AM_92920635/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-f_{1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right )=\frac {1}{\sin \left (x \right )}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sin \left (x \right )-f_{1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right ) \sin \left (x \right )-1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1+f_{1} \left (y-\ln \left (\sin \left (x \right )\right )+\ln \left (\cos \left (x \right )+1\right )\right ) \sin \left (x \right )}{\sin \left (x \right )} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4 `, `-> Computing symmetries using: way = 5 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] 1st order, trying the canonical coordinates of the invariance group -> Calling odsolve with the ODE`, diff(y(x), x) = (cos(x)^2+sin(x)^2+cos(x))/(sin(x)*(cos(x)+1)), y(x)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful <- 1st order, canonical coordinates successful <- symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] successful`
✓ Solution by Maple
Time used: 0.235 (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)
\[ y \left (x \right ) = -\ln \left (\csc \left (x \right )+\cot \left (x \right )\right )+\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.855 (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 ] \]