73.24.4 problem 34.5 (d)
Internal
problem
ID
[15605]
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
:
Thursday, March 13, 2025 at 06:12:30 AM
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: 0.022 (sec). Leaf size: 244
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);
\[
y = \left (1+\frac {{\mathrm e}^{4} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+4 \,{\mathrm e}^{4}\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left ({\mathrm e}^{2}+12 \,{\mathrm e}^{4}+\frac {33 \,{\mathrm e}^{6}}{2}+\frac {{\mathrm e}^{10}}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left ({\mathrm e}^{2}+\frac {53 \,{\mathrm e}^{4}}{2}+98 \,{\mathrm e}^{6}+79 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+\frac {5 \,{\mathrm e}^{12}}{2}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y \left (2\right )+\left (x -2-\frac {4 \,{\mathrm e}^{2} \left (x -2\right )^{2}}{{\mathrm e}^{4}-1}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {31 \,{\mathrm e}^{2}}{2}-\frac {{\mathrm e}^{6}}{2}-4\right ) \left (x -2\right )^{3}}{3 \left ({\mathrm e}^{4}-1\right )^{2}}+\frac {\left (-47 \,{\mathrm e}^{2}-65 \,{\mathrm e}^{4}-{\mathrm e}^{6}-\frac {7 \,{\mathrm e}^{8}}{2}-\frac {7}{2}\right ) {\mathrm e}^{2} \left (x -2\right )^{4}}{3 \left ({\mathrm e}^{4}-1\right )^{3}}-\frac {2 \,{\mathrm e}^{2} \left (-\frac {205 \,{\mathrm e}^{2}}{2}-\frac {1537 \,{\mathrm e}^{4}}{4}-\frac {1249 \,{\mathrm e}^{6}}{4}-\frac {85 \,{\mathrm e}^{8}}{4}-17 \,{\mathrm e}^{10}+\frac {{\mathrm e}^{12}}{4}-\frac {{\mathrm e}^{14}}{4}-\frac {11}{4}\right ) \left (x -2\right )^{5}}{15 \left ({\mathrm e}^{4}-1\right )^{4}}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right )
\]
✓ 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)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sinh {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sinh ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} e^{x + 2}}{6 \sinh {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sinh {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sinh {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (- \frac {\left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} e^{x + 2}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{4} e^{2 x + 4}}{24 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} e^{x + 2}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} e^{x + 2}}{2 \sinh {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]