41.1.8 problem Ex. 8(i), page 258
Internal
problem
ID
[6821]
Book
:
A
treatise
on
Differential
Equations
by
A.
R.
Forsyth.
6th
edition.
1929.
Macmillan
Co.
ltd.
New
York,
reprinted
1956
Section
:
Chapter
VI.
Note
I.
Integration
of
linear
equations
in
series
by
the
method
of
Frobenius.
page
243
Problem
number
:
Ex.
8(i),
page
258
Date
solved
:
Sunday, March 30, 2025 at 11:23:42 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+4 \left (x +a \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 407
Order:=6;
ode:=x^2*diff(diff(y(x),x),x)+4*(x+a)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \sqrt {x}\, \left (c_1 \,x^{-\frac {\sqrt {1-16 a}}{2}} \left (1+4 \frac {1}{-1+\sqrt {1-16 a}} x +8 \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right )} x^{2}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right )} x^{4}+\frac {128}{15} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right ) \left (-5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {\sqrt {1-16 a}}{2}} \left (1-4 \frac {1}{1+\sqrt {1-16 a}} x +8 \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right )} x^{2}-\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right )} x^{4}-\frac {128}{15} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right ) \left (5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )
\]
✓ Mathematica. Time used: 0.006 (sec). Leaf size: 1356
ode=x^2*D[y[x],{x,2}]+4*(x+a)*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")
a = symbols("a")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (4*a + 4*x)*y(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