Internal problem ID [9132]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 797.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Bernoulli]
\[ \boxed {y^{\prime }-\frac {y \left (-1-\cosh \left (\frac {x +1}{x -1}\right ) x +\cosh \left (\frac {x +1}{x -1}\right ) x^{2} y-x^{2} \cosh \left (\frac {x +1}{x -1}\right )+\cosh \left (\frac {x +1}{x -1}\right ) x^{3} y\right )}{x}=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 223
dsolve(diff(y(x),x) = y(x)*(-1-cosh((x+1)/(x-1))*x+cosh((x+1)/(x-1))*x^2*y(x)-cosh((x+1)/(x-1))*x^2+cosh((x+1)/(x-1))*x^3*y(x))/x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {{\mathrm e}^{\frac {x +1}{x -1}} x^{2}}{4}-x \,{\mathrm e}^{\frac {x +1}{x -1}}+\frac {5 \,{\mathrm e}^{\frac {x +1}{x -1}}}{4}-3 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )+\operatorname {Ei}_{1}\left (\frac {2}{x -1}\right ) {\mathrm e}^{-1}-\frac {{\mathrm e}^{-\frac {x +1}{x -1}} x^{2}}{4}+\frac {{\mathrm e}^{-\frac {x +1}{x -1}}}{4}}}{x \left (c_{1} +\int -{\mathrm e}^{-\frac {{\mathrm e}^{\frac {x +1}{x -1}} x^{2}}{4}-x \,{\mathrm e}^{\frac {x +1}{x -1}}+\frac {5 \,{\mathrm e}^{\frac {x +1}{x -1}}}{4}-3 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )+\operatorname {Ei}_{1}\left (\frac {2}{x -1}\right ) {\mathrm e}^{-1}-\frac {{\mathrm e}^{-\frac {x +1}{x -1}} x^{2}}{4}+\frac {{\mathrm e}^{-\frac {x +1}{x -1}}}{4}} \left (x +1\right ) \cosh \left (\frac {x +1}{x -1}\right )d x \right )} \]
✓ Solution by Mathematica
Time used: 2.638 (sec). Leaf size: 166
DSolve[y'[x] == (y[x]*(-1 - x*Cosh[(1 + x)/(-1 + x)] - x^2*Cosh[(1 + x)/(-1 + x)] + x^2*Cosh[(1 + x)/(-1 + x)]*y[x] + x^3*Cosh[(1 + x)/(-1 + x)]*y[x]))/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\exp \left (\frac {4 \left (3 e^2-1\right ) \text {Chi}\left (\frac {2}{x-1}\right )+4 \left (1+3 e^2\right ) \text {Shi}\left (\frac {2}{x-1}\right )+e^{-\frac {2}{x-1}}}{4 e}\right )}{x \left (\exp \left (\frac {4 \left (3 e^2-1\right ) \text {Chi}\left (\frac {2}{x-1}\right )+4 \left (1+3 e^2\right ) \text {Shi}\left (\frac {2}{x-1}\right )+e^{-\frac {2}{x-1}}}{4 e}\right )+c_1 \exp \left (\frac {1}{4} e^{\frac {x+1}{x-1}} \left (\left (e^{-\frac {2 (x+1)}{x-1}}+1\right ) x^2+4 x-5\right )\right )\right )} y(x)\to 0 \end{align*}