73.26.11 problem 36.2 (k)
Internal
problem
ID
[15680]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
36.
The
big
theorem
on
the
the
Frobenius
method.
Additional
Exercises.
page
739
Problem
number
:
36.2
(k)
Date
solved
:
Monday, March 31, 2025 at 01:44:54 PM
CAS
classification
:
[[_2nd_order, _exact, _linear, _homogeneous]]
\begin{align*} \left (x -3\right ) y^{\prime \prime }+\left (x -3\right ) y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 3 \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 62
Order:=6;
ode:=(x-3)*diff(diff(y(x),x),x)+(x-3)*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x),type='series',x=3);
\[
y = c_1 \left (x -3\right ) \left (1-\left (x -3\right )+\frac {1}{2} \left (x -3\right )^{2}-\frac {1}{6} \left (x -3\right )^{3}+\frac {1}{24} \left (x -3\right )^{4}-\frac {1}{120} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+c_2 \left (\ln \left (x -3\right ) \left (-\left (x -3\right )+\left (x -3\right )^{2}-\frac {1}{2} \left (x -3\right )^{3}+\frac {1}{6} \left (x -3\right )^{4}-\frac {1}{24} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )+\left (1-\left (x -3\right )+\frac {1}{4} \left (x -3\right )^{3}-\frac {5}{36} \left (x -3\right )^{4}+\frac {13}{288} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.02 (sec). Leaf size: 105
ode=(x-3)*D[y[x],{x,2}]+(x-3)*D[y[x],x]+y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
\[
y(x)\to c_2 \left (\frac {1}{24} (x-3)^5-\frac {1}{6} (x-3)^4+\frac {1}{2} (x-3)^3-(x-3)^2+x-3\right )+c_1 \left (\frac {1}{36} \left (-11 (x-3)^4+27 (x-3)^3-36 (x-3)^2+36\right )+\frac {1}{6} \left ((x-3)^3-3 (x-3)^2+6 (x-3)-6\right ) (x-3) \log (x-3)\right )
\]
✓ Sympy. Time used: 0.844 (sec). Leaf size: 34
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x - 3)*Derivative(y(x), x) + (x - 3)*Derivative(y(x), (x, 2)) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=3,n=6)
\[
y{\left (x \right )} = C_{1} \left (x - 3\right ) \left (- x + \frac {\left (x - 3\right )^{4}}{24} - \frac {\left (x - 3\right )^{3}}{6} + \frac {\left (x - 3\right )^{2}}{2} + 4\right ) + O\left (x^{6}\right )
\]