64.15.14 problem 14

Internal problem ID [13520]
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 : 14
Date solved : Monday, March 31, 2025 at 08:00:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{4}+x \right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 29

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+(x^4+x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 x \left (1-\frac {1}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2-\frac {2}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 30

ode=x^2*D[y[x],{x,2}]+(x^4+x)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x-\frac {x^4}{15}\right )+c_1 \left (\frac {x^2}{3}+\frac {1}{x}\right ) \]

✓ Sympy. Time used: 0.902 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x**4 + x)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x \left (1 - \frac {x^{3}}{15}\right ) + \frac {C_{1} \left (- \frac {x^{6}}{36} + \frac {x^{3}}{3} + 1\right )}{x} + O\left (x^{6}\right ) \]