60.2.221 problem 797
Internal
problem
ID
[10808]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
797
Date
solved
:
Monday, January 27, 2025 at 10:02:29 PM
CAS
classification
:
[_Bernoulli]
\begin{align*} y^{\prime }&=\frac {y \left (-1-\cosh \left (\frac {1+x}{x -1}\right ) x +\cosh \left (\frac {1+x}{x -1}\right ) x^{2} y-\cosh \left (\frac {1+x}{x -1}\right ) x^{2}+\cosh \left (\frac {1+x}{x -1}\right ) x^{3} y\right )}{x} \end{align*}
✓ Solution by Maple
Time used: 0.138 (sec). Leaf size: 171
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)
\[
y = -\frac {{\mathrm e}^{\frac {\left (-x^{2}+1\right ) {\mathrm e}^{\frac {-x -1}{x -1}}}{4}+\frac {\left (-x^{2}-4 x +5\right ) {\mathrm e}^{\frac {x +1}{x -1}}}{4}+\operatorname {Ei}_{1}\left (\frac {2}{x -1}\right ) {\mathrm e}^{-1}-3 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )}}{x \left (-c_{1} +\int {\mathrm e}^{\frac {\left (-x^{2}+1\right ) {\mathrm e}^{\frac {-x -1}{x -1}}}{4}+\frac {\left (-x^{2}-4 x +5\right ) {\mathrm e}^{\frac {x +1}{x -1}}}{4}+\operatorname {Ei}_{1}\left (\frac {2}{x -1}\right ) {\mathrm e}^{-1}-3 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-\frac {2}{x -1}\right )} \left (x +1\right ) \cosh \left (\frac {x +1}{x -1}\right )d x \right )}
\]
✓ Solution by Mathematica
Time used: 2.675 (sec). Leaf size: 216
DSolve[D[y[x],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 (\int _1^x\left (-\cosh \left (\frac {K[1]+1}{K[1]-1}\right ) (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right )}{-\int _1^x\exp \left (\int _1^{K[2]}\left (-\cosh \left (\frac {K[1]+1}{K[1]-1}\right ) (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right ) \cosh \left (\frac {K[2]+1}{K[2]-1}\right ) K[2] (K[2]+1)dK[2]+c_1} \\
y(x)\to 0 \\
y(x)\to -\frac {\exp \left (\int _1^x\left (-\cosh \left (\frac {K[1]+1}{K[1]-1}\right ) (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right )}{\int _1^x\exp \left (\int _1^{K[2]}\left (-\cosh \left (\frac {K[1]+1}{K[1]-1}\right ) (K[1]+1)-\frac {1}{K[1]}\right )dK[1]\right ) \cosh \left (\frac {K[2]+1}{K[2]-1}\right ) K[2] (K[2]+1)dK[2]} \\
\end{align*}