48.5.8 problem 8

Internal problem ID [9909]
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 : 8
Date solved : Tuesday, September 30, 2025 at 06:44:17 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*} 0 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 40

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=0);
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 71

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,0,7}]
 

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

✓ Sympy. Time used: 0.372 (sec). Leaf size: 42

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=0,n=8)
 

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