9.8.22 problem problem 22

Internal problem ID [1087]
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 22
Date solved : Saturday, March 29, 2025 at 10:38:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+6 x \right ) y^{\prime \prime }+\left (3 x +9\right ) y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -3 \end{align*}

With initial conditions

\begin{align*} y \left (-3\right )&=1\\ y^{\prime }\left (-3\right )&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

Order:=6; 
ode:=(x^2+6*x)*diff(diff(y(x),x),x)+(3*x+9)*diff(y(x),x)-3*y(x) = 0; 
ic:=y(-3) = 1, D(y)(-3) = 0; 
dsolve([ode,ic],y(x),type='series',x=-3);
 

\[ y = 1-\frac {1}{6} \left (x +3\right )^{2}-\frac {5}{648} \left (x +3\right )^{4}+\operatorname {O}\left (\left (x +3\right )^{6}\right ) \]

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

ode=(x^2+6*x)*D[y[x],{x,2}]+(3*x+9)*D[y[x],x]-3*y[x]==0; 
ic={y[-3]==1,Derivative[1][y][-3 ]==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-3,5}]
 

\[ y(x)\to -\frac {5}{648} (x+3)^4-\frac {1}{6} (x+3)^2+1 \]

✓ Sympy. Time used: 0.873 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x + 9)*Derivative(y(x), x) + (x**2 + 6*x)*Derivative(y(x), (x, 2)) - 3*y(x),0) 
ics = {y(-3): 1, Subs(Derivative(y(x), x), x, -3): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-3,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {5 \left (x + 3\right )^{4}}{648} - \frac {\left (x + 3\right )^{2}}{6} + 1\right ) + C_{1} \left (x + 3\right ) + O\left (x^{6}\right ) \]