14.26.20 problem 14

Internal problem ID [4045]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 07:00:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (x^{2}+6\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 32

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

\[ y = \frac {c_1 \left (1+\frac {1}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) x +c_2 \left (1+\frac {3}{2} x^{2}+\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 33

ode=x^2*D[y[x],{x,2}]+x*(6+x^2)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {x}{8}+\frac {3}{2 x}\right )+c_2 \left (\frac {1}{x^2}+\frac {1}{3}\right ) \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \frac {C_{2} \left (\frac {x^{2}}{3} + 1\right )}{x^{2}} + \frac {C_{1} \left (\frac {x^{8}}{4480} - \frac {x^{6}}{240} + \frac {x^{4}}{8} + \frac {3 x^{2}}{2} + 1\right )}{x^{3}} + O\left (x^{6}\right ) \]