9.8.21 problem problem 21

Internal problem ID [1086]
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 21
Date solved : Saturday, March 29, 2025 at 10:38:33 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (4 x^{2}+16 x +17\right ) y^{\prime \prime }&=8 y \end{align*}

Using series method with expansion around

\begin{align*} -2 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

Order:=6; 
ode:=(4*x^2+16*x+17)*diff(diff(y(x),x),x) = 8*y(x); 
ic:=y(-2) = 1, D(y)(-2) = 0; 
dsolve([ode,ic],y(x),type='series',x=-2);
 

\[ y = 4 x^{2}+16 x +17 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 12

ode=(4*x^2+16*x+17)*D[y[x],{x,2}]==8*y[x]; 
ic={y[-2]==1,Derivative[1][y][-2]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-2,5}]
 

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

✓ Sympy. Time used: 0.794 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x**2 + 16*x + 17)*Derivative(y(x), (x, 2)) - 8*y(x),0) 
ics = {y(-2): 1, Subs(Derivative(y(x), x), x, -2): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-2,n=6)
 

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