7.15.9 problem 9

Internal problem ID [465]
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 : 9
Date solved : Saturday, March 29, 2025 at 04:54:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (1-x \right ) y^{\prime \prime }+x y^{\prime }+x^{2} 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:=(1-x)*diff(diff(y(x),x),x)+x*diff(y(x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{12} x^{4}-\frac {1}{20} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}-\frac {1}{12} x^{4}-\frac {3}{40} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

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

ode=(1-x)*D[y[x],{x,2}]+x*D[y[x],x]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{20}-\frac {x^4}{12}+1\right )+c_2 \left (-\frac {3 x^5}{40}-\frac {x^4}{12}-\frac {x^3}{6}+x\right ) \]

✓ Sympy. Time used: 0.795 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + x*Derivative(y(x), x) + (1 - x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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