54.8.1 problem 1
Internal
problem
ID
[8663]
Book
:
Elementary
differential
equations.
Rainville,
Bedient,
Bedient.
Prentice
Hall.
NJ.
8th
edition.
1997.
Section
:
CHAPTER
18.
Power
series
solutions.
18.11
Many-Term
Recurrence
Relations.
Exercises
page
391
Problem
number
:
1
Date
solved
:
Sunday, March 30, 2025 at 01:22:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (x^{3}+x +1\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:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(x^3+x+1)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{4} x^{2}-\frac {5}{36} x^{3}+\frac {41}{576} x^{4}-\frac {37}{2880} x^{5}+\frac {437}{103680} x^{6}-\frac {7817}{5080320} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {3}{4} x^{2}+\frac {19}{108} x^{3}-\frac {593}{3456} x^{4}+\frac {3629}{86400} x^{5}-\frac {7733}{1036800} x^{6}+\frac {485257}{118540800} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2}{x}
\]
✓ Mathematica. Time used: 0.009 (sec). Leaf size: 164
ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+(1+x+x^3)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to \frac {c_1 \left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right )}{x}+c_2 \left (\frac {\frac {485257 x^7}{118540800}-\frac {7733 x^6}{1036800}+\frac {3629 x^5}{86400}-\frac {593 x^4}{3456}+\frac {19 x^3}{108}-\frac {3 x^2}{4}+2 x}{x}+\frac {\left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)}{x}\right )
\]
✓ Sympy. Time used: 0.894 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + (x**3 + x + 1)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
\[
y{\left (x \right )} = \frac {C_{1} \left (\frac {14617 x^{8}}{65028096} - \frac {7817 x^{7}}{5080320} + \frac {437 x^{6}}{103680} - \frac {37 x^{5}}{2880} + \frac {41 x^{4}}{576} - \frac {5 x^{3}}{36} + \frac {x^{2}}{4} - x + 1\right )}{x} + O\left (x^{8}\right )
\]