50.19.2 problem 1(b)
Internal
problem
ID
[8110]
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.
Section
4.4.
REGULAR
SINGULAR
POINTS.
Page
175
Problem
number
:
1(b)
Date
solved
:
Sunday, March 30, 2025 at 12:45:37 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+2 y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 367
Order:=8;
ode:=x^2*(x^2-1)*diff(diff(y(x),x),x)-x*(1-x)*diff(y(x),x)+2*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \,x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}}{-1+2 \sqrt {2}} x +\frac {\sqrt {2}}{-5+3 \sqrt {2}} x^{2}+\frac {-8+6 \sqrt {2}}{57 \sqrt {2}-81} x^{3}+\frac {-49 \sqrt {2}+69}{1104-780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}-414}{6108 \sqrt {2}-8640} x^{5}+\frac {-2757 \sqrt {2}+3898}{114408-80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}-109686}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (2 \sqrt {2}-3\right ) \left (-2+\sqrt {2}\right ) \left (-5+2 \sqrt {2}\right ) \left (-3+\sqrt {2}\right ) \left (-7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{\sqrt {2}} \left (1+\frac {\sqrt {2}}{1+2 \sqrt {2}} x +\frac {\sqrt {2}}{5+3 \sqrt {2}} x^{2}+\frac {6 \sqrt {2}+8}{57 \sqrt {2}+81} x^{3}+\frac {49 \sqrt {2}+69}{1104+780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}+414}{6108 \sqrt {2}+8640} x^{5}+\frac {2757 \sqrt {2}+3898}{114408+80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}+109686}{\left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right ) \left (3+\sqrt {2}\right ) \left (7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.015 (sec). Leaf size: 10352
ode=x^2*(x^2-1)*D[y[x],{x,2}]-x*(1-x)*D[y[x],x]+2*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
Too large to display
✓ Sympy. Time used: 0.995 (sec). Leaf size: 22
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*(x**2 - 1)*Derivative(y(x), (x, 2)) - x*(1 - x)*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)
\[
y{\left (x \right )} = C_{2} x^{\sqrt {2}} + \frac {C_{1}}{x^{\sqrt {2}}} + O\left (x^{8}\right )
\]