7.15.12 problem 12

Internal problem ID [468]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 12
Date solved : Saturday, March 29, 2025 at 04:54:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -2\right )^{3} y^{\prime \prime }+3 \left (x -2\right )^{2} y^{\prime }+x^{3} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

Order:=6; 
ode:=(x-2)^3*diff(diff(y(x),x),x)+3*(x-2)^2*diff(y(x),x)+x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {x^{5}}{160}\right ) y \left (0\right )+\left (x +\frac {3}{4} x^{2}+\frac {1}{2} x^{3}+\frac {5}{16} x^{4}+\frac {3}{16} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 49

ode=(x-2)^3*D[y[x],{x,2}]+3*(x-2)^2*D[y[x],x]+x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{160}+1\right )+c_2 \left (\frac {3 x^5}{16}+\frac {5 x^4}{16}+\frac {x^3}{2}+\frac {3 x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.942 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3*y(x) + (x - 2)**3*Derivative(y(x), (x, 2)) + 3*(x - 2)**2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = O\left (1\right ) \]