64.15.26 problem 26

Internal problem ID [13524]
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 : 26
Date solved : Wednesday, March 05, 2025 at 10:03:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\left (x^{2}-3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 44

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

\[ y = \frac {c_{1} x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+(x^2-3)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {\left (x^2+8\right )^2}{64 x}-\frac {1}{16} x^3 \log (x)\right )+c_2 \left (\frac {x^7}{384}-\frac {x^5}{12}+x^3\right ) \]

✓ Sympy. Time used: 0.800 (sec). Leaf size: 17

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 - 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{3} \left (1 - \frac {x^{2}}{12}\right ) + O\left (x^{6}\right ) \]