54.8.6 problem 8

Internal problem ID [8668]
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 : 8
Date solved : Sunday, March 30, 2025 at 01:23:00 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*} 0 \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 40

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=0);
 

\[ y = \left (\frac {1}{2} x -\frac {1}{8} x^{2}-\frac {1}{48} x^{3}-\frac {1}{192} x^{4}-\frac {1}{640} x^{5}-\frac {1}{1920} x^{6}-\frac {1}{5376} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 +\left (1-\frac {1}{2} x +\operatorname {O}\left (x^{8}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 75

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,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]

✓ Sympy. Time used: 0.961 (sec). Leaf size: 42

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=0,n=8)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{7}}{198450} + \frac {x^{6}}{8100} - \frac {x^{5}}{450} + \frac {x^{4}}{36} - \frac {2 x^{3}}{9} + x^{2} - 2 x + 1\right ) + O\left (x^{8}\right ) \]