Internal problem ID [9183]
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)]`]]
\[ \boxed {y^{\prime }-f_{1} \left (y-\ln \left (\sinh \left (x \right )\right )\right )=\frac {\cosh \left (x \right )}{\sinh \left (x \right )}} \]
✓ 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]
\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\text {F1}(K[2]-\log (\sinh (x))) \int _1^x\left (\frac {(\coth (K[1])+\text {F1}(K[2]-\log (\sinh (K[1])))) \text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))^2}-\frac {\text {F1}'(K[2]-\log (\sinh (K[1])))}{\text {F1}(K[2]-\log (\sinh (K[1])))}\right )dK[1]-1}{\text {F1}(K[2]-\log (\sinh (x)))}dK[2]+\int _1^x-\frac {\coth (K[1])+\text {F1}(y(x)-\log (\sinh (K[1])))}{\text {F1}(y(x)-\log (\sinh (K[1])))}dK[1]=c_1,y(x)\right ] \]