Internal problem ID [8428]
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.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(y)]]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {\cosh \relax (x )}{\sinh \relax (x )}-f_{1}\left (y-\ln \left (\sinh \relax (x )\right )\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 27
dsolve(diff(y(x),x) = 1/sinh(x)*cosh(x)+_F1(y(x)-ln(sinh(x))),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \relax (x )}\frac {1}{f_{1}\left (\textit {\_a} -\ln \left (\sinh \relax (x )\right )\right )}d \textit {\_a} -x -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.336 (sec). Leaf size: 148
DSolve[y'[x] == Coth[x] + F1[-Log[Sinh[x]] + y[x]],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}(K[2]-\log (\sinh (x))) \int _1^x\left (\frac {(\coth (K[1])+\text {F1}(K[2]-\log (\sinh (K[1])))) \text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))^2}-\frac {\text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))}\right )dK[1]-1}{\text {F1}(K[2]-\log (\sinh (x)))}dK[2]+\int _1^x-\frac {\coth (K[1])+\text {F1}(y(x)-\log (\sinh (K[1])))}{\text {F1}(y(x)-\log (\sinh (K[1])))}dK[1]=c_1,y(x)\right ] \]