78.19.18 problem 7 (a)

Internal problem ID [18366]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 7 (a)
Date solved : Monday, March 31, 2025 at 05:26:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {n y^{\prime }}{x^{2}}+\frac {q y}{x^{3}}&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=diff(diff(y(x),x),x)+n/x^2*diff(y(x),x)+q/x^3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.756 (sec). Leaf size: 790

ode=D[y[x],{x,2}]+n/x^2*D[y[x],x]+q/x^3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
q = symbols("q") 
y = Function("y") 
ode = Eq(n*Derivative(y(x), x)/x**2 + q*y(x)/x**3 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE n*Derivative(y(x), x)/x**2 + q*y(x)/x**3 + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular