64.15.7 problem 7

Internal problem ID [13505]
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 : 7
Date solved : Wednesday, March 05, 2025 at 10:02:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}+\frac {8}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 35

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+(x^2+8/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+(x^2+8/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.862 (sec). Leaf size: 42

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 + 8/9)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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