7.15.14 problem 14

Internal problem ID [470]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 14
Date solved : Saturday, March 29, 2025 at 04:54:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-9\right )^{2} y^{\prime \prime }+\left (x^{2}+9\right ) y^{\prime }+\left (x^{2}+4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

Order:=6; 
ode:=(x^2-9)^2*diff(diff(y(x),x),x)+(x^2+9)*diff(y(x),x)+(x^2+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {2}{81} x^{2}+\frac {2}{2187} x^{3}-\frac {49}{26244} x^{4}+\frac {463}{3542940} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{18} x^{2}-\frac {1}{162} x^{3}-\frac {47}{17496} x^{4}-\frac {697}{787320} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 70

ode=(x^2-9)^2*D[y[x],{x,2}]+(x^2+9)*D[y[x],x]+(x^2+4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {463 x^5}{3542940}-\frac {49 x^4}{26244}+\frac {2 x^3}{2187}-\frac {2 x^2}{81}+1\right )+c_2 \left (-\frac {697 x^5}{787320}-\frac {47 x^4}{17496}-\frac {x^3}{162}-\frac {x^2}{18}+x\right ) \]

✓ Sympy. Time used: 1.128 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 - 9)**2*Derivative(y(x), (x, 2)) + (x**2 + 4)*y(x) + (x**2 + 9)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = - \frac {x^{4} r{\left (3 \right )}}{36} + \frac {7 x^{5} r{\left (3 \right )}}{108} + C_{2} \left (\frac {253 x^{5}}{3542940} - \frac {145 x^{4}}{78732} - \frac {2 x^{2}}{81} + 1\right ) + C_{1} x \left (- \frac {191 x^{4}}{393660} - \frac {25 x^{3}}{8748} - \frac {x}{18} + 1\right ) + O\left (x^{6}\right ) \]