54.8.7 problem 9

Internal problem ID [8669]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 9
Date solved : Sunday, March 30, 2025 at 01:23:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 44

Order:=8; 
ode:=x*(x-2)^2*diff(diff(y(x),x),x)-2*(x-2)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = \left (\left (-\frac {1}{2} \left (x -2\right )+\frac {1}{8} \left (x -2\right )^{2}-\frac {1}{24} \left (x -2\right )^{3}+\frac {1}{64} \left (x -2\right )^{4}-\frac {1}{160} \left (x -2\right )^{5}+\frac {1}{384} \left (x -2\right )^{6}-\frac {1}{896} \left (x -2\right )^{7}+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right ) c_2 +\left (1+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right ) \left (c_2 \ln \left (x -2\right )+c_1 \right )\right ) \left (x -2\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 90

ode=x*(x-2)^2*D[y[x],{x,2}]-2*(x-2)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,7}]
 

\[ y(x)\to c_1 (x-2)+c_2 \left (\left (-\frac {1}{896} (x-2)^7+\frac {1}{384} (x-2)^6-\frac {1}{160} (x-2)^5+\frac {1}{64} (x-2)^4-\frac {1}{24} (x-2)^3+\frac {1}{8} (x-2)^2+\frac {2-x}{2}\right ) (x-2)+(x-2) \log (x-2)\right ) \]

✓ Sympy. Time used: 0.905 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x - 2)**2*Derivative(y(x), (x, 2)) - (2*x - 4)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=2,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (x - 2\right ) + O\left (x^{8}\right ) \]