73.24.4 problem 34.5 (d)
Internal
problem
ID
[15613]
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
(d)
Date
solved
:
Monday, March 31, 2025 at 01:42:34 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-{\mathrm e}^{x} y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 2 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 196
Order:=6;
ode:=sinh(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-exp(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=2);
\[
y = \left (1+\frac {{\mathrm e}^{2} \operatorname {csch}\left (2\right ) \left (x -2\right )^{2}}{2}+\frac {\left (-8 \,{\mathrm e}^{2}-2\right ) \left (x -2\right )^{3}}{12 \sinh \left (2\right )^{2}}+\frac {\left ({\mathrm e}^{-2}+\frac {33 \,{\mathrm e}^{2}}{2}+\frac {{\mathrm e}^{6}}{2}+12\right ) \left (x -2\right )^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (-392-106 \,{\mathrm e}^{-2}-316 \,{\mathrm e}^{2}-4 \,{\mathrm e}^{-4}-12 \,{\mathrm e}^{4}-10 \,{\mathrm e}^{6}\right ) \left (x -2\right )^{5}}{480 \sinh \left (2\right )^{4}}\right ) y \left (2\right )+\left (x -2-2 \,\operatorname {csch}\left (2\right ) \left (x -2\right )^{2}+\frac {\left ({\mathrm e}^{4}+8 \,{\mathrm e}^{-2}+31\right ) \left (x -2\right )^{3}}{12 \sinh \left (2\right )^{2}}+\frac {\left (-65-7 \cosh \left (4\right )-{\mathrm e}^{2}-47 \,{\mathrm e}^{-2}\right ) \left (x -2\right )^{4}}{24 \sinh \left (2\right )^{3}}+\frac {\left (410 \,{\mathrm e}^{-4}+{\mathrm e}^{8}+85 \,{\mathrm e}^{2}+68 \,{\mathrm e}^{4}+1537 \,{\mathrm e}^{-2}+11 \,{\mathrm e}^{-6}-{\mathrm e}^{6}+1249\right ) \left (x -2\right )^{5}}{480 \sinh \left (2\right )^{4}}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 931
ode=Sinh[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Exp[x]*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,5}]
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Sympy. Time used: 5.234 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x) - y(x)*exp(x) + sinh(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sinh {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sinh ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} e^{x + 2}}{6 \sinh {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sinh {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sinh {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{4} e^{2 x + 4}}{24 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} e^{x + 2}}{2 \sinh {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]