54.7.13 problem 13
Internal
problem
ID
[8661]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
18.9
Indicial
Equation
with
Difference
of
Roots
a
Positive
Integer:
Logarithmic
Case.
Exercises
page
384
Problem
number
:
13
Date
solved
:
Wednesday, March 05, 2025 at 06:11:49 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 9 x^{2} y^{\prime \prime }-15 x y^{\prime }+7 \left (x +1\right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 74
Order:=8;
ode:=9*x^2*diff(diff(y(x),x),x)-15*x*diff(y(x),x)+7*(1+x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \left (x^{2} \left (1-\frac {7}{27} x +\frac {49}{1944} x^{2}-\frac {343}{262440} x^{3}+\frac {2401}{56687040} x^{4}-\frac {2401}{2550916800} x^{5}+\frac {16807}{1101996057600} x^{6}-\frac {16807}{89261680665600} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{1} +c_{2} \left (\ln \left (x \right ) \left (\frac {49}{81} x^{2}-\frac {343}{2187} x^{3}+\frac {2401}{157464} x^{4}-\frac {16807}{21257640} x^{5}+\frac {117649}{4591650240} x^{6}-\frac {117649}{206624260800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-2-\frac {14}{9} x +\frac {1372}{6561} x^{3}-\frac {60025}{1889568} x^{4}+\frac {2638699}{1275458400} x^{5}-\frac {10706059}{137749507200} x^{6}+\frac {11916163}{6198727824000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right )\right ) x^{{1}/{3}}
\]
✓ Mathematica. Time used: 0.056 (sec). Leaf size: 141
ode=9*x^2*D[y[x],{x,2}]-15*x*D[y[x],x]+7*(x+1)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_2 \left (\frac {16807 x^{25/3}}{1101996057600}-\frac {2401 x^{22/3}}{2550916800}+\frac {2401 x^{19/3}}{56687040}-\frac {343 x^{16/3}}{262440}+\frac {49 x^{13/3}}{1944}-\frac {7 x^{10/3}}{27}+x^{7/3}\right )+c_1 \left (\frac {\sqrt [3]{x} \left (6235397 x^6-169717086 x^5+2713009950 x^4-19803722400 x^3+20832487200 x^2+107138505600 x+137749507200\right )}{137749507200}-\frac {49 x^{7/3} \left (2401 x^4-74088 x^3+1428840 x^2-14696640 x+56687040\right ) \log (x)}{9183300480}\right )
\]
✓ Sympy. Time used: 0.808 (sec). Leaf size: 39
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) - 15*x*Derivative(y(x), x) + (7*x + 7)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{1} x^{\frac {7}{3}} \left (\frac {2401 x^{4}}{56687040} - \frac {343 x^{3}}{262440} + \frac {49 x^{2}}{1944} - \frac {7 x}{27} + 1\right ) + O\left (x^{8}\right )
\]