40.17.11 problem 21
Internal
problem
ID
[6812]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
26.
Integration
in
series
(singular
points).
Supplemetary
problems.
Page
218
Problem
number
:
21
Date
solved
:
Wednesday, March 05, 2025 at 02:46:20 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{\prime \prime }+\left (x^{2}+x \right ) y^{\prime }-y&=0 \end{align*}
Using series method with expansion around
\begin{align*} \infty \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 112
Order:=6;
ode:=x^3*diff(diff(y(x),x),x)+(x^2+x)*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x),type='series',x=infinity);
\[
y = \frac {7200 \left (O\left (\frac {1}{x^{6}}\right ) x^{5}+x^{5}+x^{4}+\frac {x^{3}}{2}+\frac {x^{2}}{6}+\frac {x}{24}+\frac {1}{120}\right ) c_2 \ln \left (\frac {1}{x}\right )+7200 x^{5} \left (c_1 +c_2 \right ) O\left (\frac {1}{x^{6}}\right )+7200 c_1 \,x^{5}+\left (7200 c_1 -7200 c_2 \right ) x^{4}+\left (3600 c_1 -5400 c_2 \right ) x^{3}+\left (1200 c_1 -2200 c_2 \right ) x^{2}+\left (300 c_1 -625 c_2 \right ) x +60 c_1 -137 c_2}{7200 x^{5}}
\]
✓ Mathematica. Time used: 0.007 (sec). Leaf size: 124
ode=x^3*D[y[x],{x,2}]+(x^2+x)*D[y[x],x]-y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,Infinity,5}]
\[
y(x)\to c_1 \left (\frac {1}{120 x^5}+\frac {1}{24 x^4}+\frac {1}{6 x^3}+\frac {1}{2 x^2}+\frac {1}{x}+1\right )+c_2 \left (-\frac {137}{7200 x^5}-\frac {\log (x)}{120 x^5}-\frac {25}{288 x^4}-\frac {\log (x)}{24 x^4}-\frac {11}{36 x^3}-\frac {\log (x)}{6 x^3}-\frac {3}{4 x^2}-\frac {\log (x)}{2 x^2}-\frac {1}{x}-\frac {\log (x)}{x}-\log (x)\right )
\]
✓ Sympy. Time used: 1.673 (sec). Leaf size: 175
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3*Derivative(y(x), (x, 2)) + (x**2 + x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=inf,n=6)
\[
y{\left (x \right )} = C_{3} \left (- C_{2} + x - \frac {\left (C_{2} - x\right )^{2}}{2 C_{2}} - \frac {\left (C_{2} - x\right )^{3}}{3 C_{2}^{2}} - \frac {\left (C_{2} - x\right )^{2}}{2 C_{2}^{2}} - \frac {\left (C_{2} - x\right )^{4}}{4 C_{2}^{3}} - \frac {5 \left (C_{2} - x\right )^{3}}{6 C_{2}^{3}} - \frac {13 \left (C_{2} - x\right )^{4}}{12 C_{2}^{4}} - \frac {\left (C_{2} - x\right )^{3}}{6 C_{2}^{4}} - \frac {11 \left (C_{2} - x\right )^{4}}{24 C_{2}^{5}} - \frac {\left (C_{2} - x\right )^{4}}{24 C_{2}^{6}}\right ) + C_{1} \left (1 + \frac {\left (C_{2} - x\right )^{2}}{2 C_{2}^{3}} + \frac {2 \left (C_{2} - x\right )^{3}}{3 C_{2}^{4}} + \frac {3 \left (C_{2} - x\right )^{4}}{4 C_{2}^{5}} + \frac {\left (C_{2} - x\right )^{3}}{6 C_{2}^{5}} + \frac {5 \left (C_{2} - x\right )^{4}}{12 C_{2}^{6}} + \frac {\left (C_{2} - x\right )^{4}}{24 C_{2}^{7}}\right ) + O\left (x^{6}\right )
\]