64.15.5 problem 5

Internal problem ID [13511]
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 : 5
Date solved : Monday, March 31, 2025 at 08:00:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 33

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

\[ y = \frac {c_1 \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 48

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

\[ y(x)\to c_1 x \left (\frac {x^4}{616}-\frac {x^2}{14}+1\right )+\frac {c_2 \left (\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.866 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{616} - \frac {x^{2}}{14} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{40} - \frac {x^{2}}{2} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]