40.2.1 problem 2

Internal problem ID [8590]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:39:29 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 49

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

\[ y = \left (1+\frac {1}{8} x^{2}+\frac {1}{48} x^{3}-\frac {1}{480} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{240} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_1 \left (-\frac {x^5}{480}+\frac {x^3}{48}+\frac {x^2}{8}+1\right )+c_2 \left (\frac {x^5}{240}-\frac {x^3}{24}-\frac {x^2}{4}+x\right ) \]

✓ Sympy. Time used: 0.292 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{3}}{48} + \frac {x^{2}}{8} + 1\right ) + C_{1} x \left (- \frac {x^{2}}{24} - \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]