54.8.2 problem 2

Internal problem ID [8664]
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 : 2
Date solved : Sunday, March 30, 2025 at 01:22:52 PM
CAS classification : [_Jacobi]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 52

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {1}{2} x -\frac {9}{40} x^{2}-\frac {149}{1680} x^{3}-\frac {661}{13440} x^{4}-\frac {16171}{492800} x^{5}-\frac {5530601}{230630400} x^{6}-\frac {299137703}{16144128000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (1-2 x -\frac {1}{6} x^{2}+\frac {1}{15} x^{3}+\frac {37}{840} x^{4}+\frac {527}{18900} x^{5}+\frac {16309}{831600} x^{6}+\frac {14339}{970200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {299137703 x^7}{16144128000}-\frac {5530601 x^6}{230630400}-\frac {16171 x^5}{492800}-\frac {661 x^4}{13440}-\frac {149 x^3}{1680}-\frac {9 x^2}{40}-\frac {x}{2}+1\right )+c_2 \left (\frac {14339 x^7}{970200}+\frac {16309 x^6}{831600}+\frac {527 x^5}{18900}+\frac {37 x^4}{840}+\frac {x^3}{15}-\frac {x^2}{6}-2 x+1\right ) \]

✓ Sympy. Time used: 1.089 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*(1 - x)*Derivative(y(x), (x, 2)) + (1 - 2*x)*Derivative(y(x), x) + (x + 2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} + C_{1} + O\left (x^{8}\right ) \]