64.15.20 problem 20

Internal problem ID [13518]
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 : 20
Date solved : Wednesday, March 05, 2025 at 10:03:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.061 (sec). Leaf size: 44

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

\[ y = \frac {c_{1} x^{4} \left (1-\frac {1}{4} x^{2}+\frac {5}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-9\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \left (\frac {5 x^7}{128}-\frac {x^5}{4}+x^3\right )+c_1 \left (\frac {1}{16} x^3 \log (x)-\frac {x^4+16 x^2-64}{64 x}\right ) \]

✓ Sympy. Time used: 0.771 (sec). Leaf size: 17

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

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