54.5.2 problem 2
Internal
problem
ID
[8617]
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
:
2
Date
solved
:
Sunday, March 30, 2025 at 01:21:23 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 x^{2} y^{\prime \prime }+\left (1-2 x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.029 (sec). Leaf size: 56
Order:=8;
ode:=4*x^2*diff(diff(y(x),x),x)+(-2*x+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{2} x +\frac {1}{16} x^{2}+\frac {1}{288} x^{3}+\frac {1}{9216} x^{4}+\frac {1}{460800} x^{5}+\frac {1}{33177600} x^{6}+\frac {1}{3251404800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{16} x^{2}-\frac {11}{864} x^{3}-\frac {25}{55296} x^{4}-\frac {137}{13824000} x^{5}-\frac {49}{331776000} x^{6}-\frac {121}{75866112000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right )
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 174
ode=4*x^2*D[y[x],{x,2}]+(1-2*x)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {121 x^7}{75866112000}-\frac {49 x^6}{331776000}-\frac {137 x^5}{13824000}-\frac {25 x^4}{55296}-\frac {11 x^3}{864}-\frac {3 x^2}{16}-x\right )+\sqrt {x} \left (\frac {x^7}{3251404800}+\frac {x^6}{33177600}+\frac {x^5}{460800}+\frac {x^4}{9216}+\frac {x^3}{288}+\frac {x^2}{16}+\frac {x}{2}+1\right ) \log (x)\right )
\]
✓ Sympy. Time used: 0.843 (sec). Leaf size: 42
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + (1 - 2*x)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {x^{6}}{33177600} + \frac {x^{5}}{460800} + \frac {x^{4}}{9216} + \frac {x^{3}}{288} + \frac {x^{2}}{16} + \frac {x}{2} + 1\right ) + O\left (x^{8}\right )
\]