6.12.1 problem 1

Internal problem ID [1855]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 05:20:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

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

\[ y = \left (5 x^{4}-3 x^{2}+1\right ) y \left (0\right )+\left (3 x^{5}-2 x^{3}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

ode=(1+x^2)*D[y[x],{x,2}]+6*x*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (3 x^5-2 x^3+x\right )+c_1 \left (5 x^4-3 x^2+1\right ) \]

✓ Sympy. Time used: 0.247 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{2} \left (5 x^{4} - 3 x^{2} + 1\right ) + C_{1} x \left (1 - 2 x^{2}\right ) + O\left (x^{6}\right ) \]