73.24.7 problem 34.5 (g)
Internal
problem
ID
[15616]
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
:
Monday, March 31, 2025 at 01:43:02 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }+\frac {\left (1+{\mathrm e}^{x}\right ) y}{1-{\mathrm e}^{x}}&=0 \end{align*}
Using series method with expansion around
\begin{align*} 3 \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 159
Order:=6;
ode:=diff(diff(y(x),x),x)+(exp(x)+1)/(-exp(x)+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=3);
\[
y = \left (1+\frac {\coth \left (\frac {3}{2}\right ) \left (x -3\right )^{2}}{2}-\frac {{\mathrm e}^{3} \left (x -3\right )^{3}}{3 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left (-1+{\mathrm e}^{9}+3 \,{\mathrm e}^{6}+{\mathrm e}^{3}\right ) \left (x -3\right )^{4}}{24 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left (6 \,{\mathrm e}^{3}-8 \,{\mathrm e}^{6}-10 \,{\mathrm e}^{9}\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) y \left (3\right )+\left (x -3+\frac {\left ({\mathrm e}^{6}-1\right ) \left (x -3\right )^{3}}{6 \left (-1+{\mathrm e}^{3}\right )^{2}}+\frac {\left (-{\mathrm e}^{6}+{\mathrm e}^{3}\right ) \left (x -3\right )^{4}}{6 \left (-1+{\mathrm e}^{3}\right )^{3}}+\frac {\left (1+{\mathrm e}^{12}+6 \,{\mathrm e}^{9}-2 \,{\mathrm e}^{6}-6 \,{\mathrm e}^{3}\right ) \left (x -3\right )^{5}}{120 \left (-1+{\mathrm e}^{3}\right )^{4}}\right ) y^{\prime }\left (3\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.005 (sec). Leaf size: 275
ode=D[y[x],{x,2}]+(1+Exp[x])/(1-Exp[x])*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
\[
y(x)\to c_1 \left (-\frac {\left (e^3+4 e^6+e^9\right ) (x-3)^5}{60 \left (e^3-1\right )^4}-\frac {e^3 \left (1+e^3\right ) (x-3)^5}{60 \left (e^3-1\right )^3}+\frac {e^3 \left (-1-e^3\right ) (x-3)^5}{20 \left (e^3-1\right )^3}-\frac {\left (-e^3-e^6\right ) (x-3)^4}{12 \left (e^3-1\right )^3}+\frac {\left (-1-e^3\right )^2 (x-3)^4}{24 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^3}{3 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^2}{2 \left (e^3-1\right )}+1\right )+c_2 \left (-\frac {\left (-e^3-e^6\right ) (x-3)^5}{20 \left (e^3-1\right )^3}+\frac {\left (1+e^3\right )^2 (x-3)^5}{120 \left (e^3-1\right )^2}-\frac {e^3 (x-3)^4}{6 \left (e^3-1\right )^2}+\frac {\left (1+e^3\right ) (x-3)^3}{6 \left (e^3-1\right )}+x-3\right )
\]
✓ Sympy. Time used: 1.233 (sec). Leaf size: 109
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), (x, 2)) + (exp(x) + 1)*y(x)/(1 - exp(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=3,n=6)
\[
y{\left (x \right )} = C_{2} \left (\frac {\left (x - 3\right )^{4} \left (e^{x + 3} + 1\right )^{2}}{24 \left (- 2 e^{x + 3} + e^{2 x + 6} + 1\right )} + \frac {\left (x - 3\right )^{2} e^{x + 3}}{2 \left (e^{x + 3} - 1\right )} + \frac {\left (x - 3\right )^{2}}{2 \left (e^{x + 3} - 1\right )} + 1\right ) + C_{1} \left (x + \frac {\left (x - 3\right )^{3} e^{x + 3}}{6 \left (e^{x + 3} - 1\right )} + \frac {\left (x - 3\right )^{3}}{6 \left (e^{x + 3} - 1\right )} - 3\right ) + O\left (x^{6}\right )
\]