84.34.9 problem 20.22
Internal
problem
ID
[22336]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
20.
Regular
singular
points
and
the
method
of
Frobenius.
Supplementary
problems
Problem
number
:
20.22
Date
solved
:
Thursday, October 02, 2025 at 08:37:42 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+\left (x^{2}-3 x \right ) y^{\prime }+\left (3 x -1\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 367
Order:=6;
ode:=x^2*diff(diff(y(x),x),x)+(x^2-3*x)*diff(y(x),x)+(3*x-1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = x^{2} \left (c_1 \,x^{-\sqrt {5}} \left (1+\frac {-5+\sqrt {5}}{1-2 \sqrt {5}} x +\frac {\left (-6+\sqrt {5}\right ) \left (-5+\sqrt {5}\right )}{44-12 \sqrt {5}} x^{2}-\frac {\left (-7+\sqrt {5}\right ) \left (-6+\sqrt {5}\right ) \left (-5+\sqrt {5}\right )}{372 \sqrt {5}-756} x^{3}+\frac {\left (-8+\sqrt {5}\right ) \left (-7+\sqrt {5}\right ) \left (-6+\sqrt {5}\right ) \left (-5+\sqrt {5}\right )}{26976-12000 \sqrt {5}} x^{4}-\frac {1}{480} \frac {\left (-9+\sqrt {5}\right ) \left (-8+\sqrt {5}\right ) \left (-7+\sqrt {5}\right ) \left (-6+\sqrt {5}\right ) \left (-5+\sqrt {5}\right )}{\left (-1+2 \sqrt {5}\right ) \left (\sqrt {5}-1\right ) \left (-3+2 \sqrt {5}\right ) \left (-2+\sqrt {5}\right ) \left (-5+2 \sqrt {5}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\sqrt {5}} \left (1+\frac {-5-\sqrt {5}}{1+2 \sqrt {5}} x +\frac {\left (6+\sqrt {5}\right ) \left (5+\sqrt {5}\right )}{44+12 \sqrt {5}} x^{2}-\frac {\left (7+\sqrt {5}\right ) \left (6+\sqrt {5}\right ) \left (5+\sqrt {5}\right )}{372 \sqrt {5}+756} x^{3}+\frac {\left (8+\sqrt {5}\right ) \left (7+\sqrt {5}\right ) \left (6+\sqrt {5}\right ) \left (5+\sqrt {5}\right )}{26976+12000 \sqrt {5}} x^{4}-\frac {1}{480} \frac {\left (9+\sqrt {5}\right ) \left (8+\sqrt {5}\right ) \left (7+\sqrt {5}\right ) \left (6+\sqrt {5}\right ) \left (5+\sqrt {5}\right )}{\left (1+2 \sqrt {5}\right ) \left (\sqrt {5}+1\right ) \left (3+2 \sqrt {5}\right ) \left (2+\sqrt {5}\right ) \left (5+2 \sqrt {5}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 1244
ode=x^2*D[y[x],{x,2}]+(x^2-3*x)*D[y[x],x]+(3*x-1)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (3*x - 1)*y(x) + (x**2 - 3*x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
ValueError : Expected Expr or iterable but got None