12.15.13 problem 9
Internal
problem
ID
[2011]
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
:
9
Date
solved
:
Tuesday, March 04, 2025 at 01:48:05 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+x +4\right ) y^{\prime }+\left (3 x^{2}+5 x +1\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 56
Order:=8;
ode:=4*x^2*diff(diff(y(x),x),x)+2*x*(x^2+x+4)*diff(y(x),x)+(3*x^2+5*x+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{4} x^{2}+\frac {1}{18} x^{3}-\frac {37}{1152} x^{4}-\frac {17}{28800} x^{5}+\frac {593}{259200} x^{6}-\frac {1913}{12700800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {3}{2} x -\frac {13}{16} x^{2}+\frac {1}{54} x^{3}+\frac {1103}{13824} x^{4}-\frac {19507}{1728000} x^{5}-\frac {98531}{20736000} x^{6}+\frac {982189}{889056000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{\sqrt {x}}
\]
✓ Mathematica. Time used: 0.011 (sec). Leaf size: 172
ode=4*x^2*D[y[x],{x,2}]+2*x*(4+x+x^2)*D[y[x],x]+(1+5*x+3*x^2)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to \frac {c_1 \left (-\frac {1913 x^7}{12700800}+\frac {593 x^6}{259200}-\frac {17 x^5}{28800}-\frac {37 x^4}{1152}+\frac {x^3}{18}+\frac {x^2}{4}-x+1\right )}{\sqrt {x}}+c_2 \left (\frac {\frac {982189 x^7}{889056000}-\frac {98531 x^6}{20736000}-\frac {19507 x^5}{1728000}+\frac {1103 x^4}{13824}+\frac {x^3}{54}-\frac {13 x^2}{16}+\frac {3 x}{2}}{\sqrt {x}}+\frac {\left (-\frac {1913 x^7}{12700800}+\frac {593 x^6}{259200}-\frac {17 x^5}{28800}-\frac {37 x^4}{1152}+\frac {x^3}{18}+\frac {x^2}{4}-x+1\right ) \log (x)}{\sqrt {x}}\right )
\]
✓ Sympy. Time used: 1.105 (sec). Leaf size: 53
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 2*x*(x**2 + x + 4)*Derivative(y(x), x) + (3*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)
\[
y{\left (x \right )} = \frac {C_{1} \left (- \frac {1913 x^{7}}{12700800} + \frac {593 x^{6}}{259200} - \frac {17 x^{5}}{28800} - \frac {37 x^{4}}{1152} + \frac {x^{3}}{18} + \frac {x^{2}}{4} - x + 1\right )}{\sqrt {x}} + O\left (x^{8}\right )
\]