64.15.9 problem 9

Internal problem ID [13507]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 10:02:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-1/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {x^{{2}/{3}} \left (1-\frac {3}{16} x^{2}+\frac {9}{896} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} +\left (1-\frac {3}{8} x^{2}+\frac {9}{320} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1}}{x^{{1}/{3}}} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {9 x^4}{896}-\frac {3 x^2}{16}+1\right )+\frac {c_2 \left (\frac {9 x^4}{320}-\frac {3 x^2}{8}+1\right )}{\sqrt [3]{x}} \]

✓ Sympy. Time used: 0.881 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 1/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {9 x^{4}}{896} - \frac {3 x^{2}}{16} + 1\right ) + \frac {C_{1} \left (\frac {9 x^{4}}{320} - \frac {3 x^{2}}{8} + 1\right )}{\sqrt [3]{x}} + O\left (x^{6}\right ) \]