44.22.12 problem 2(d)
Internal
problem
ID
[9441]
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.
(A)
Drill
Exercises
.
Page
194
Problem
number
:
2(d)
Date
solved
:
Tuesday, September 30, 2025 at 06:18:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 x^{2} y^{\prime \prime }+4 x^{2} y^{\prime }+2 y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 55
Order:=8;
ode:=4*x^2*diff(diff(y(x),x),x)+4*x^2*diff(y(x),x)+2*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \,x^{\frac {1}{2}-\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}+\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}-\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}-\frac {977 i}{48307200}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{\frac {1}{2}+\frac {i}{2}} \left (1-\frac {1}{2} x +\left (\frac {7}{40}-\frac {i}{40}\right ) x^{2}+\left (-\frac {11}{240}+\frac {i}{80}\right ) x^{3}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{4}+\left (-\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{5}+\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{6}+\left (-\frac {30269}{1014451200}+\frac {977 i}{48307200}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.014 (sec). Leaf size: 226
ode=4*x^2*D[y[x],{x,2}]+4*x^2*D[y[x],x]+2*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \left (\left (\frac {3421}{14492160}-\frac {223 i}{1610240}\right ) x^{\frac {13}{2}+\frac {i}{2}}-\left (\frac {53}{32640}-\frac {13 i}{16320}\right ) x^{\frac {11}{2}+\frac {i}{2}}+\left (\frac {31}{3264}-\frac {i}{272}\right ) x^{\frac {9}{2}+\frac {i}{2}}-\left (\frac {11}{240}-\frac {i}{80}\right ) x^{\frac {7}{2}+\frac {i}{2}}+\left (\frac {7}{40}-\frac {i}{40}\right ) x^{\frac {5}{2}+\frac {i}{2}}-\frac {1}{2} x^{\frac {3}{2}+\frac {i}{2}}+x^{\frac {1}{2}+\frac {i}{2}}\right )+c_2 \left (\left (\frac {3421}{14492160}+\frac {223 i}{1610240}\right ) x^{\frac {13}{2}-\frac {i}{2}}-\left (\frac {53}{32640}+\frac {13 i}{16320}\right ) x^{\frac {11}{2}-\frac {i}{2}}+\left (\frac {31}{3264}+\frac {i}{272}\right ) x^{\frac {9}{2}-\frac {i}{2}}-\left (\frac {11}{240}+\frac {i}{80}\right ) x^{\frac {7}{2}-\frac {i}{2}}+\left (\frac {7}{40}+\frac {i}{40}\right ) x^{\frac {5}{2}-\frac {i}{2}}-\frac {1}{2} x^{\frac {3}{2}-\frac {i}{2}}+x^{\frac {1}{2}-\frac {i}{2}}\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + 2*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
ValueError : Expected Expr or iterable but got None