54.9.6 problem 6

Internal problem ID [8676]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 6
Date solved : Sunday, March 30, 2025 at 01:23:16 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-10 x y^{\prime }-18 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 49

Order:=8; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-10*x*diff(y(x),x)-18*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (70 x^{6}+30 x^{4}+9 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {14}{3} x^{3}+\frac {63}{5} x^{5}+\frac {132}{5} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

ode=(1-x^2)*D[y[x],{x,2}]-10*x*D[y[x],x]-18*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (\frac {132 x^7}{5}+\frac {63 x^5}{5}+\frac {14 x^3}{3}+x\right )+c_1 \left (70 x^6+30 x^4+9 x^2+1\right ) \]

✓ Sympy. Time used: 0.825 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*x*Derivative(y(x), x) + (1 - x**2)*Derivative(y(x), (x, 2)) - 18*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (70 x^{6} + 30 x^{4} + 9 x^{2} + 1\right ) + C_{1} x \left (\frac {63 x^{4}}{5} + \frac {14 x^{2}}{3} + 1\right ) + O\left (x^{8}\right ) \]