9.7.23 problem problem 23

Internal problem ID [1064]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.1 Introduction and Review of power series. Page 615
Problem number : problem 23
Date solved : Thursday, March 13, 2025 at 03:53:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 228

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (c_2 \,x^{\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {i \sqrt {3}+3}{8 i \sqrt {3}+16} x^{2}+\frac {-i \sqrt {3}-5}{48 i \sqrt {3}+96} x^{3}+\frac {1}{384} \frac {\left (i \sqrt {3}+5\right ) \left (i \sqrt {3}+7\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{4}-\frac {1}{3840} \frac {\left (i \sqrt {3}+7\right ) \left (i \sqrt {3}+9\right )}{\left (i \sqrt {3}+4\right ) \left (i \sqrt {3}+2\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \,x^{-\frac {i \sqrt {3}}{2}} \left (1-\frac {1}{2} x +\frac {\sqrt {3}+3 i}{8 \sqrt {3}+16 i} x^{2}+\frac {-\sqrt {3}-5 i}{48 \sqrt {3}+96 i} x^{3}+\frac {3 i \sqrt {3}-8}{576 i \sqrt {3}-480} x^{4}-\frac {1}{3840} \frac {\left (\sqrt {3}+7 i\right ) \left (\sqrt {3}+9 i\right )}{\left (\sqrt {3}+4 i\right ) \left (\sqrt {3}+2 i\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 886

ode=x^2*D[y[x],{x,2}]+x^2*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : Expected Expr or iterable but got None