54.9.15 problem 15
Internal
problem
ID
[8685]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
Miscellaneous
Exercises.
page
394
Problem
number
:
15
Date
solved
:
Sunday, March 30, 2025 at 01:23:31 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 x^{2} y^{\prime \prime }-x^{2} y^{\prime }+y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 54
Order:=8;
ode:=4*x^2*diff(diff(y(x),x),x)-x^2*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{8} x +\frac {3}{256} x^{2}+\frac {5}{6144} x^{3}+\frac {35}{786432} x^{4}+\frac {21}{10485760} x^{5}+\frac {77}{1006632960} x^{6}+\frac {143}{56371445760} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{256} x^{2}-\frac {1}{2048} x^{3}-\frac {19}{524288} x^{4}-\frac {25}{12582912} x^{5}-\frac {317}{3623878656} x^{6}-\frac {469}{144955146240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \sqrt {x}
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 171
ode=4*x^2*D[y[x],{x,2}]-x^2*D[y[x],x]+y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \sqrt {x} \left (\frac {143 x^7}{56371445760}+\frac {77 x^6}{1006632960}+\frac {21 x^5}{10485760}+\frac {35 x^4}{786432}+\frac {5 x^3}{6144}+\frac {3 x^2}{256}+\frac {x}{8}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {469 x^7}{144955146240}-\frac {317 x^6}{3623878656}-\frac {25 x^5}{12582912}-\frac {19 x^4}{524288}-\frac {x^3}{2048}-\frac {x^2}{256}\right )+\sqrt {x} \left (\frac {143 x^7}{56371445760}+\frac {77 x^6}{1006632960}+\frac {21 x^5}{10485760}+\frac {35 x^4}{786432}+\frac {5 x^3}{6144}+\frac {3 x^2}{256}+\frac {x}{8}+1\right ) \log (x)\right )
\]
✓ Sympy. Time used: 0.808 (sec). Leaf size: 51
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {77 x^{6}}{1006632960} + \frac {21 x^{5}}{10485760} + \frac {35 x^{4}}{786432} + \frac {5 x^{3}}{6144} + \frac {3 x^{2}}{256} + \frac {x}{8} + 1\right ) + O\left (x^{8}\right )
\]