6.13.2 problem 2

Internal problem ID [1893]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:20:59 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (2 x^{2}+x +1\right ) y^{\prime \prime }+\left (2+8 x \right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

Order:=6; 
ode:=(2*x^2+x+1)*diff(diff(y(x),x),x)+(2+8*x)*diff(y(x),x)+4*y(x) = 0; 
ic:=[y(0) = -1, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 23

ode=(1+x+2*x^2)*D[y[x],{x,2}]+(2+8*x)*D[y[x],x]+4*y[x]==0; 
ic={y[0]==-1,Derivative[1][y][0] ==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.263 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((8*x + 2)*Derivative(y(x), x) + (2*x**2 + x + 1)*Derivative(y(x), (x, 2)) + 4*y(x),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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