9.8.4 problem problem 4

Internal problem ID [1069]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 4
Date solved : Saturday, March 29, 2025 at 10:38:10 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+4 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)+4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (3 x^{4}-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {5}{3} x^{3}+\frac {7}{3} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 60

ode=(x^2+1)*D[y[x],{x,2}]+6*D[y[x],x]+4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (4 x^5-5 x^4+4 x^3-2 x^2+1\right )+c_2 \left (\frac {77 x^5}{15}-\frac {13 x^4}{2}+\frac {16 x^3}{3}-3 x^2+x\right ) \]

✓ Sympy. Time used: 0.747 (sec). Leaf size: 31

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)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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