73.24.7 problem 34.5 (g)
Internal
problem
ID
[15687]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
34.
Power
series
solutions
II:
Generalization
and
theory.
Additional
Exercises.
page
678
Problem
number
:
34.5
(g)
Date
solved
:
Tuesday, January 28, 2025 at 08:05:18 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\frac {\left ({\mathrm e}^{x}+1\right ) y}{1-{\mathrm e}^{x}}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 3 \end{align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 168
Order:=6;
dsolve(diff(y(x),x$2)+(1+exp(x))/(1-exp(x))*y(x)=0,y(x),type='series',x=3);
\[
y = \left (1+\frac {\left ({\mathrm e}^{3}+1\right ) \left (x -3\right )^{2}}{-2+2 \,{\mathrm e}^{3}}-\frac {{\mathrm e}^{3} \left (x -3\right )^{3}}{3 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left ({\mathrm e}^{3}+3 \,{\mathrm e}^{6}+{\mathrm e}^{9}-1\right ) \left (x -3\right )^{4}}{24 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left (6 \,{\mathrm e}^{3}-8 \,{\mathrm e}^{6}-10 \,{\mathrm e}^{9}\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) y \left (3\right )+\left (x -3+\frac {\left ({\mathrm e}^{6}-1\right ) \left (x -3\right )^{3}}{6 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left ({\mathrm e}^{3}-{\mathrm e}^{6}\right ) \left (x -3\right )^{4}}{6 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left (-6 \,{\mathrm e}^{3}-2 \,{\mathrm e}^{6}+6 \,{\mathrm e}^{9}+{\mathrm e}^{12}+1\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) y^{\prime }\left (3\right )+O\left (x^{6}\right )
\]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 275
AsymptoticDSolveValue[D[y[x],{x,2}]+(1+Exp[x])/(1-Exp[x])*y[x]==0,y[x],{x,3,"6"-1}]
\[
y(x)\to c_1 \left (-\frac {\left (e^3+4 e^6+e^9\right ) (x-3)^5}{60 \left (e^3-1\right )^4}-\frac {e^3 \left (1+e^3\right ) (x-3)^5}{60 \left (e^3-1\right )^3}+\frac {e^3 \left (-1-e^3\right ) (x-3)^5}{20 \left (e^3-1\right )^3}-\frac {\left (-e^3-e^6\right ) (x-3)^4}{12 \left (e^3-1\right )^3}+\frac {\left (-1-e^3\right )^2 (x-3)^4}{24 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^3}{3 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^2}{2 \left (e^3-1\right )}+1\right )+c_2 \left (-\frac {\left (-e^3-e^6\right ) (x-3)^5}{20 \left (e^3-1\right )^3}+\frac {\left (1+e^3\right )^2 (x-3)^5}{120 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^4}{6 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^3}{6 \left (e^3-1\right )}+x-3\right )
\]