54.5.9 problem 9

Internal problem ID [8624]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 9
Date solved : Sunday, March 30, 2025 at 01:21:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (x -2\right ) y^{\prime \prime }+2 \left (x -1\right ) y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 42

Order:=8; 
ode:=x*(x-2)*diff(diff(y(x),x),x)+2*(x-1)*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=2);
 

\[ y = \left (-\frac {5}{2} \left (x -2\right )-\frac {3}{8} \left (x -2\right )^{2}+\frac {1}{12} \left (x -2\right )^{3}-\frac {5}{192} \left (x -2\right )^{4}+\frac {3}{320} \left (x -2\right )^{5}-\frac {7}{1920} \left (x -2\right )^{6}+\frac {1}{672} \left (x -2\right )^{7}+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right ) c_2 +\left (1+\left (x -2\right )+\operatorname {O}\left (\left (x -2\right )^{8}\right )\right ) \left (c_2 \ln \left (x -2\right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 90

ode=x*(x-2)*D[y[x],{x,2}]+2*(x-1)*D[y[x],x]-2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,2,7}]
 

\[ y(x)\to c_1 (x-1)+c_2 \left (\frac {1}{672} (x-2)^7-\frac {7 (x-2)^6}{1920}+\frac {3}{320} (x-2)^5-\frac {5}{192} (x-2)^4+\frac {1}{12} (x-2)^3-\frac {3}{8} (x-2)^2-2 (x-2)+\frac {2-x}{2}+(x-1) \log (x-2)\right ) \]

✓ Sympy. Time used: 0.981 (sec). Leaf size: 7

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x - 2)*Derivative(y(x), (x, 2)) + (2*x - 2)*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)
 

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