73.24.5 problem 34.5 (e)
Internal
problem
ID
[15606]
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
(e)
Date
solved
:
Thursday, March 13, 2025 at 06:12:35 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sinh \left (x \right ) y^{\prime \prime }+x^{2} y^{\prime }-\sin \left (x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 2 \end{align*}
✓ Maple. Time used: 0.041 (sec). Leaf size: 458
Order:=6;
ode:=sinh(x)*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-sin(x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=2);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 0.004 (sec). Leaf size: 934
ode=Sinh[x]*D[y[x],{x,2}]+x^2*D[y[x],x]-Sin[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: 6.241 (sec). Leaf size: 228
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), x) - y(x)*sin(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)
\[
y{\left (x \right )} = C_{2} \left (x - \frac {\left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4}}{12 \sinh {\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4}}{\sinh ^{2}{\left (x + 2 \right )}} - \frac {8 \left (x - 2\right )^{4}}{3 \sinh ^{3}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{3} \sin {\left (x + 2 \right )}}{6 \sinh {\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3}}{3 \sinh {\left (x + 2 \right )}} + \frac {8 \left (x - 2\right )^{3}}{3 \sinh ^{2}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{2}}{\sinh {\left (x + 2 \right )}} - 2\right ) + C_{1} \left (\frac {\left (x - 2\right )^{4} \sin ^{2}{\left (x + 2 \right )}}{24 \sinh ^{2}{\left (x + 2 \right )}} - \frac {\left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {2 \left (x - 2\right )^{4} \sin {\left (x + 2 \right )}}{3 \sinh ^{3}{\left (x + 2 \right )}} - \frac {2 \left (x - 2\right )^{3} \sin {\left (x + 2 \right )}}{3 \sinh ^{2}{\left (x + 2 \right )}} + \frac {\left (x - 2\right )^{2} \sin {\left (x + 2 \right )}}{2 \sinh {\left (x + 2 \right )}} + 1\right ) + O\left (x^{6}\right )
\]