46.2.8 problem 9

Internal problem ID [7311]
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 : 9
Date solved : Wednesday, March 05, 2025 at 04:23:06 AM
CAS classification : [_Jacobi]

\begin{align*} 2 x \left (x -1\right ) y^{\prime \prime }-\left (x +1\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: 26

Order:=6; 
ode:=2*x*(x-1)*diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} \sqrt {x}\, \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 18

ode=2*x*(x-1)*D[y[x],{x,2}]-(x+1)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x}+c_2 (x+1) \]

✓ Sympy. Time used: 1.041 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(x - 1)*Derivative(y(x), (x, 2)) - (x + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + C_{1} + O\left (x^{6}\right ) \]