54.9.9 problem 9

Internal problem ID [8679]
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 : 9
Date solved : Sunday, March 30, 2025 at 01:23:21 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (-x^{2}+1\right ) y^{\prime \prime }-\left (x^{2}+7\right ) y^{\prime }+4 x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 32

Order:=8; 
ode:=x*(-x^2+1)*diff(diff(y(x),x),x)-(x^2+7)*diff(y(x),x)+4*x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{8} \left (1+3 x^{2}+6 x^{4}+10 x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (-203212800-67737600 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 38

ode=x*(1-x^2)*D[y[x],{x,2}]-(7+x^2)*D[y[x],x]+4*x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{3}+1\right )+c_2 \left (10 x^{14}+6 x^{12}+3 x^{10}+x^8\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(1 - x**2)*Derivative(y(x), (x, 2)) + 4*x*y(x) - (x**2 + 7)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : Expected Expr or iterable but got None