7.13.6 problem 6

Internal problem ID [405]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.1 (Introduction). Problems at page 206
Problem number : 6
Date solved : Saturday, March 29, 2025 at 04:52:56 PM
CAS classification : [_separable]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 37

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

\[ y = \left (1+\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {1}{8} x^{3}+\frac {1}{16} x^{4}+\frac {1}{32} x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 41

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

\[ y(x)\to c_1 \left (\frac {x^5}{32}+\frac {x^4}{16}+\frac {x^3}{8}+\frac {x^2}{4}+\frac {x}{2}+1\right ) \]

✓ Sympy. Time used: 0.686 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{1} + \frac {C_{1} x}{2} + \frac {C_{1} x^{2}}{4} + \frac {C_{1} x^{3}}{8} + \frac {C_{1} x^{4}}{16} + \frac {C_{1} x^{5}}{32} + O\left (x^{6}\right ) \]