50.23.1 problem 1(a)
Internal
problem
ID
[8164]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
4.
Power
Series
Solutions
and
Special
Functions.
Problems
for
review
and
discovert.
(B)
Challenge
Problems
.
Page
194
Problem
number
:
1(a)
Date
solved
:
Sunday, March 30, 2025 at 12:47:06 PM
CAS
classification
:
[[_Emden, _Fowler]]
\begin{align*} x^{3} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} \infty \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 146
Order:=8;
ode:=x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x),type='series',x=infinity);
\[
y = \frac {1778112000 \left (O\left (\frac {1}{x^{8}}\right ) x^{7}+x^{7}-x^{6}+\frac {x^{5}}{4}-\frac {x^{4}}{36}+\frac {x^{3}}{576}-\frac {x^{2}}{14400}+\frac {x}{518400}-\frac {1}{25401600}\right ) c_2 \ln \left (\frac {1}{x}\right )+1778112000 x^{7} \left (c_1 +c_2 \right ) O\left (\frac {1}{x^{8}}\right )+1778112000 c_1 \,x^{7}+\left (-1778112000 c_1 +3556224000 c_2 \right ) x^{6}+\left (444528000 c_1 -1333584000 c_2 \right ) x^{5}+\left (-49392000 c_1 +181104000 c_2 \right ) x^{4}+\left (3087000 c_1 -12862500 c_2 \right ) x^{3}+\left (-123480 c_1 +563892 c_2 \right ) x^{2}+\left (3430 c_1 -16807 c_2 \right ) x -70 c_1 +363 c_2}{1778112000 x^{7}}
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 171
ode=x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,7}]
\[
y(x)\to c_1 \left (-\frac {1}{25401600 x^7}+\frac {1}{518400 x^6}-\frac {1}{14400 x^5}+\frac {1}{576 x^4}-\frac {1}{36 x^3}+\frac {1}{4 x^2}-\frac {1}{x}+1\right )+c_2 \left (\frac {121}{592704000 x^7}+\frac {\log (x)}{25401600 x^7}-\frac {49}{5184000 x^6}-\frac {\log (x)}{518400 x^6}+\frac {137}{432000 x^5}+\frac {\log (x)}{14400 x^5}-\frac {25}{3456 x^4}-\frac {\log (x)}{576 x^4}+\frac {11}{108 x^3}+\frac {\log (x)}{36 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{4 x^2}+\frac {2}{x}+\frac {\log (x)}{x}-\log (x)\right )
\]
✓ Sympy. Time used: 1.609 (sec). Leaf size: 231
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=inf,n=8)
\[
y{\left (x \right )} = C_{3} \left (- C_{2} + x - \frac {\left (C_{2} - x\right )^{2}}{2 C_{2}} - \frac {\left (C_{2} - x\right )^{3}}{3 C_{2}^{2}} - \frac {\left (C_{2} - x\right )^{4}}{4 C_{2}^{3}} + \frac {\left (C_{2} - x\right )^{3}}{6 C_{2}^{3}} - \frac {\left (C_{2} - x\right )^{5}}{5 C_{2}^{4}} + \frac {\left (C_{2} - x\right )^{4}}{3 C_{2}^{4}} - \frac {\left (C_{2} - x\right )^{6}}{6 C_{2}^{5}} + \frac {29 \left (C_{2} - x\right )^{5}}{60 C_{2}^{5}} + \frac {37 \left (C_{2} - x\right )^{6}}{60 C_{2}^{6}} - \frac {\left (C_{2} - x\right )^{5}}{120 C_{2}^{6}} - \frac {7 \left (C_{2} - x\right )^{6}}{240 C_{2}^{7}}\right ) + C_{1} \left (1 - \frac {\left (C_{2} - x\right )^{2}}{2 C_{2}^{3}} - \frac {2 \left (C_{2} - x\right )^{3}}{3 C_{2}^{4}} - \frac {3 \left (C_{2} - x\right )^{4}}{4 C_{2}^{5}} - \frac {4 \left (C_{2} - x\right )^{5}}{5 C_{2}^{6}} + \frac {\left (C_{2} - x\right )^{4}}{24 C_{2}^{6}} - \frac {5 \left (C_{2} - x\right )^{6}}{6 C_{2}^{7}} + \frac {7 \left (C_{2} - x\right )^{5}}{60 C_{2}^{7}} + \frac {13 \left (C_{2} - x\right )^{6}}{60 C_{2}^{8}} - \frac {\left (C_{2} - x\right )^{6}}{720 C_{2}^{9}}\right ) + O\left (x^{8}\right )
\]