Internal problem ID [9182]
Internal file name [OUTPUT/8117_Monday_June_06_2022_01_48_45_AM_51131266/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 848.
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.
\[ \boxed {y^{\prime }-f_{1} \left (y-\ln \left (\sinh \left (x \right )\right )\right )=\frac {\cosh \left (x \right )}{\sinh \left (x \right )}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \sinh \left (x \right )-f_{1} \left (y-\ln \left (\sinh \left (x \right )\right )\right ) \sinh \left (x \right )-\cosh \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cosh \left (x \right )+f_{1} \left (y-\ln \left (\sinh \left (x \right )\right )\right ) \sinh \left (x \right )}{\sinh \left (x \right )} \end {array} \]
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`[0, f__1(y-ln(sinh(x)))]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 22
dsolve(diff(y(x),x) = 1/sinh(x)*cosh(x)+_F1(y(x)-ln(sinh(x))),y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (\sinh \left (x \right )\right )+\operatorname {RootOf}\left (x -\left (\int _{}^{\textit {\_Z}}\frac {1}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.624 (sec). Leaf size: 148
DSolve[y'[x] == Coth[x] + F1[-Log[Sinh[x]] + y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}(K[2]-\log (\sinh (x))) \int _1^x\left (\frac {(\coth (K[1])+\text {F1}(K[2]-\log (\sinh (K[1])))) \text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))^2}-\frac {\text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))}\right )dK[1]-1}{\text {F1}(K[2]-\log (\sinh (x)))}dK[2]+\int _1^x-\frac {\coth (K[1])+\text {F1}(y(x)-\log (\sinh (K[1])))}{\text {F1}(y(x)-\log (\sinh (K[1])))}dK[1]=c_1,y(x)\right ] \]