54.4.7 problem 7
Internal
problem
ID
[8591]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
18.4
Indicial
Equation
with
Difference
of
Roots
Nonintegral.
Exercises
page
365
Problem
number
:
7
Date
solved
:
Sunday, March 30, 2025 at 01:20:39 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 8 x^{2} y^{\prime \prime }+10 x y^{\prime }-\left (1+x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.028 (sec). Leaf size: 56
Order:=8;
ode:=8*x^2*diff(diff(y(x),x),x)+10*x*diff(y(x),x)-(1+x)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \frac {c_2 \,x^{{3}/{4}} \left (1+\frac {1}{14} x +\frac {1}{616} x^{2}+\frac {1}{55440} x^{3}+\frac {1}{8426880} x^{4}+\frac {1}{1938182400} x^{5}+\frac {1}{627971097600} x^{6}+\frac {1}{272539456358400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_1 \left (1+\frac {1}{2} x +\frac {1}{40} x^{2}+\frac {1}{2160} x^{3}+\frac {1}{224640} x^{4}+\frac {1}{38188800} x^{5}+\frac {1}{9623577600} x^{6}+\frac {1}{3368252160000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{\sqrt {x}}
\]
✓ Mathematica. Time used: 0.008 (sec). Leaf size: 118
ode=8*x^2*D[y[x],{x,2}]+10*x*D[y[x],x]-(1+x)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to c_1 \sqrt [4]{x} \left (\frac {x^7}{272539456358400}+\frac {x^6}{627971097600}+\frac {x^5}{1938182400}+\frac {x^4}{8426880}+\frac {x^3}{55440}+\frac {x^2}{616}+\frac {x}{14}+1\right )+\frac {c_2 \left (\frac {x^7}{3368252160000}+\frac {x^6}{9623577600}+\frac {x^5}{38188800}+\frac {x^4}{224640}+\frac {x^3}{2160}+\frac {x^2}{40}+\frac {x}{2}+1\right )}{\sqrt {x}}
\]
✓ Sympy. Time used: 0.919 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(8*x**2*Derivative(y(x), (x, 2)) + 10*x*Derivative(y(x), x) - (x + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = C_{2} \sqrt [4]{x} \left (\frac {x^{6}}{627971097600} + \frac {x^{5}}{1938182400} + \frac {x^{4}}{8426880} + \frac {x^{3}}{55440} + \frac {x^{2}}{616} + \frac {x}{14} + 1\right ) + \frac {C_{1} \left (\frac {x^{7}}{3368252160000} + \frac {x^{6}}{9623577600} + \frac {x^{5}}{38188800} + \frac {x^{4}}{224640} + \frac {x^{3}}{2160} + \frac {x^{2}}{40} + \frac {x}{2} + 1\right )}{\sqrt {x}} + O\left (x^{8}\right )
\]