50.22.9 problem 2(a)
Internal
problem
ID
[8152]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
4.
Power
Series
Solutions
and
Special
Functions.
Problems
for
review
and
discovert.
(A)
Drill
Exercises
.
Page
194
Problem
number
:
2(a)
Date
solved
:
Sunday, March 30, 2025 at 12:46:49 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (x^{2}+1\right ) x^{2} y^{\prime \prime }-x y^{\prime }+\left (2+x \right ) y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 55
Order:=8;
ode:=(x^2+1)*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(x+2)*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = c_1 \,x^{1-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}+\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}-\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}+\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}+\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}-\frac {106907 i}{17808000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{1+i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {13 i}{40}\right ) x^{2}+\left (\frac {71}{520}-\frac {17 i}{520}\right ) x^{3}+\left (-\frac {31}{832}+\frac {541 i}{4160}\right ) x^{4}+\left (-\frac {1423}{20800}-\frac {7 i}{4160}\right ) x^{5}+\left (\frac {12849}{416000}-\frac {10853 i}{156000}\right ) x^{6}+\left (\frac {209609}{5088000}+\frac {106907 i}{17808000}\right ) x^{7}+\operatorname {O}\left (x^{8}\right )\right )
\]
✓ Mathematica. Time used: 0.033 (sec). Leaf size: 122
ode=(x^2+1)*x^2*D[y[x],{x,2}]-x*D[y[x],x]+(2+x)*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
\[
y(x)\to \left (\frac {1}{156000}+\frac {i}{1248000}\right ) c_2 x^{1-i} \left ((6080+10093 i) x^6-(10476-1572 i) x^5-(8220+19260 i) x^4+(21600+2400 i) x^3+(2400+50400 i) x^2-(38400+57600 i) x+(153600-19200 i)\right )-\left (\frac {1}{1248000}+\frac {i}{156000}\right ) c_1 x^{1+i} \left ((10093+6080 i) x^6+(1572-10476 i) x^5-(19260+8220 i) x^4+(2400+21600 i) x^3+(50400+2400 i) x^2-(57600+38400 i) x-(19200-153600 i)\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + (x + 2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
ValueError : Expected Expr or iterable but got None