54.5.10 problem 10
Internal
problem
ID
[8625]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
18.6.
Indicial
Equation
with
Equal
Roots.
Exercises
page
373
Problem
number
:
10
Date
solved
:
Wednesday, March 05, 2025 at 06:10:55 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 \left (x -4\right )^{2} y^{\prime \prime }+\left (x -4\right ) \left (x -8\right ) y^{\prime }+x y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 4 \end{align*}
✓ Maple. Time used: 0.013 (sec). Leaf size: 58
Order:=8;
ode:=4*(x-4)^2*diff(diff(y(x),x),x)+(x-4)*(x-8)*diff(y(x),x)+x*y(x) = 0;
dsolve(ode,y(x),type='series',x=4);
\[
y = \left (x -4\right ) \left (\left (\ln \left (x -4\right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{2} \left (x -4\right )+\frac {3}{32} \left (x -4\right )^{2}-\frac {1}{96} \left (x -4\right )^{3}+\frac {5}{6144} \left (x -4\right )^{4}-\frac {1}{20480} \left (x -4\right )^{5}+\frac {7}{2949120} \left (x -4\right )^{6}-\frac {1}{10321920} \left (x -4\right )^{7}+\operatorname {O}\left (\left (x -4\right )^{8}\right )\right )+\left (\frac {3}{4} \left (x -4\right )-\frac {13}{64} \left (x -4\right )^{2}+\frac {31}{1152} \left (x -4\right )^{3}-\frac {173}{73728} \left (x -4\right )^{4}+\frac {187}{1228800} \left (x -4\right )^{5}-\frac {463}{58982400} \left (x -4\right )^{6}+\frac {971}{2890137600} \left (x -4\right )^{7}+\operatorname {O}\left (\left (x -4\right )^{8}\right )\right ) c_{2} \right )
\]
✓ Mathematica. Time used: 0.02 (sec). Leaf size: 222
ode=4*(x-4)^2*D[y[x],{x,2}]+(x-4)*(x-8)*D[y[x],x]+x*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,4,7}]
\[
y(x)\to c_1 \left (-\frac {(x-4)^7}{10321920}+\frac {7 (x-4)^6}{2949120}-\frac {(x-4)^5}{20480}+\frac {5 (x-4)^4}{6144}-\frac {1}{96} (x-4)^3+\frac {3}{32} (x-4)^2+\frac {4-x}{2}+1\right ) (x-4)+c_2 \left ((x-4) \left (\frac {971 (x-4)^7}{2890137600}-\frac {463 (x-4)^6}{58982400}+\frac {187 (x-4)^5}{1228800}-\frac {173 (x-4)^4}{73728}+\frac {31 (x-4)^3}{1152}-\frac {13}{64} (x-4)^2+\frac {4-x}{4}+x-4\right )+\left (-\frac {(x-4)^7}{10321920}+\frac {7 (x-4)^6}{2949120}-\frac {(x-4)^5}{20480}+\frac {5 (x-4)^4}{6144}-\frac {1}{96} (x-4)^3+\frac {3}{32} (x-4)^2+\frac {4-x}{2}+1\right ) (x-4) \log (x-4)\right )
\]
✓ Sympy. Time used: 1.083 (sec). Leaf size: 54
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*y(x) + (x - 8)*(x - 4)*Derivative(y(x), x) + 4*(x - 4)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=4,n=8)
\[
y{\left (x \right )} = C_{1} \left (x - 4\right ) \left (- \frac {x}{2} + \frac {7 \left (x - 4\right )^{6}}{2949120} - \frac {\left (x - 4\right )^{5}}{20480} + \frac {5 \left (x - 4\right )^{4}}{6144} - \frac {\left (x - 4\right )^{3}}{96} + \frac {3 \left (x - 4\right )^{2}}{32} + 3\right ) + O\left (x^{8}\right )
\]