45.2.3 problem 3

Internal problem ID [7226]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 3
Date solved : Sunday, March 30, 2025 at 11:51:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

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

\[ y = \left (1-\frac {1}{81} x^{2}+\frac {1}{6561} x^{3}-\frac {289}{708588} x^{4}+\frac {304}{23914845} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{54} x^{2}-\frac {13}{2187} x^{3}-\frac {131}{236196} x^{4}-\frac {596}{1594323} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (\frac {304 x^5}{23914845}-\frac {289 x^4}{708588}+\frac {x^3}{6561}-\frac {x^2}{81}+1\right )+c_2 \left (-\frac {596 x^5}{1594323}-\frac {131 x^4}{236196}-\frac {13 x^3}{2187}-\frac {x^2}{54}+x\right ) \]

✓ Sympy. Time used: 1.138 (sec). Leaf size: 68

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

\[ y{\left (x \right )} = - \frac {x^{4} r{\left (3 \right )}}{108} + \frac {232 x^{5} r{\left (3 \right )}}{3645} + C_{2} \left (\frac {8 x^{5}}{2657205} - \frac {8 x^{4}}{19683} - \frac {x^{2}}{81} + 1\right ) + C_{1} x \left (\frac {4 x^{4}}{885735} - \frac {4 x^{3}}{6561} - \frac {x}{54} + 1\right ) + O\left (x^{6}\right ) \]