54.9.5 problem 5

Internal problem ID [8675]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 5
Date solved : Sunday, March 30, 2025 at 01:23:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (x^{2}+3\right ) y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 30

Order:=8; 
ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)+2*x*(x^2+3)*diff(y(x),x)+6*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 26

ode=x^2*(1+x^2)*D[y[x],{x,2}]+2*x*(3+x^2)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 0.978 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 2*x*(x**2 + 3)*Derivative(y(x), x) + 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

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