12.15.14 problem 10
Internal
problem
ID
[2012]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
7
Series
Solutions
of
Linear
Second
Equations.
7.6
THE
METHOD
OF
FROBENIUS
II.
Exercises
7.6.
Page
374
Problem
number
:
10
Date
solved
:
Saturday, March 29, 2025 at 11:45:47 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 56
Order:=8;
ode:=16*x^2*diff(diff(y(x),x),x)+4*x*(2*x^2+x+6)*diff(y(x),x)+(18*x^2+5*x+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x -\frac {7}{32} x^{2}+\frac {23}{384} x^{3}+\frac {145}{6144} x^{4}-\frac {881}{122880} x^{5}-\frac {4919}{2949120} x^{6}+\frac {47207}{82575360} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x +\frac {5}{64} x^{2}-\frac {157}{2304} x^{3}-\frac {841}{73728} x^{4}+\frac {65017}{7372800} x^{5}+\frac {50791}{58982400} x^{6}-\frac {953509}{1284505600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x^{{1}/{4}}}
\]
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 176
ode=16*x^2*D[y[x],{x,2}]+4*x*(6+x+2*x^2)*D[y[x],x]+(1+5*x+18*x^2)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to \frac {c_1 \left (\frac {47207 x^7}{82575360}-\frac {4919 x^6}{2949120}-\frac {881 x^5}{122880}+\frac {145 x^4}{6144}+\frac {23 x^3}{384}-\frac {7 x^2}{32}-\frac {x}{4}+1\right )}{\sqrt [4]{x}}+c_2 \left (\frac {-\frac {953509 x^7}{1284505600}+\frac {50791 x^6}{58982400}+\frac {65017 x^5}{7372800}-\frac {841 x^4}{73728}-\frac {157 x^3}{2304}+\frac {5 x^2}{64}+\frac {x}{4}}{\sqrt [4]{x}}+\frac {\left (\frac {47207 x^7}{82575360}-\frac {4919 x^6}{2949120}-\frac {881 x^5}{122880}+\frac {145 x^4}{6144}+\frac {23 x^3}{384}-\frac {7 x^2}{32}-\frac {x}{4}+1\right ) \log (x)}{\sqrt [4]{x}}\right )
\]
✓ Sympy. Time used: 1.059 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(16*x**2*Derivative(y(x), (x, 2)) + 4*x*(2*x**2 + x + 6)*Derivative(y(x), x) + (18*x**2 + 5*x + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = \frac {C_{1} \left (\frac {47207 x^{7}}{82575360} - \frac {4919 x^{6}}{2949120} - \frac {881 x^{5}}{122880} + \frac {145 x^{4}}{6144} + \frac {23 x^{3}}{384} - \frac {7 x^{2}}{32} - \frac {x}{4} + 1\right )}{\sqrt [4]{x}} + O\left (x^{8}\right )
\]