7.15.8 problem 8

Internal problem ID [464]
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 : 8
Date solved : Saturday, March 29, 2025 at 04:54:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{3}+6 x^{2}\right ) y^{\prime \prime }+21 x y^{\prime }+9 \left (x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 48

Order:=6; 
ode:=(2*x^3+6*x^2)*diff(diff(y(x),x),x)+21*x*diff(y(x),x)+9*(x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{7}/{2}} \left (1+\frac {1}{54} x -\frac {325}{2376} x^{2}+\frac {1361}{185328} x^{3}+\frac {81733}{13343616} x^{4}-\frac {153623}{151227648} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+\frac {8}{5} x +\frac {47}{30} x^{2}+\frac {62}{27} x^{3}-\frac {47}{40} x^{4}-\frac {62}{135} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 88

ode=(6*x^2+2*x^3)*D[y[x],{x,2}]+21*x*D[y[x],x]+9*(x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {153623 x^5}{151227648}+\frac {81733 x^4}{13343616}+\frac {1361 x^3}{185328}-\frac {325 x^2}{2376}+\frac {x}{54}+1\right )+\frac {c_2 \left (-\frac {62 x^5}{135}-\frac {47 x^4}{40}+\frac {62 x^3}{27}+\frac {47 x^2}{30}+\frac {8 x}{5}+1\right )}{x^3} \]

✓ Sympy. Time used: 1.071 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(21*x*Derivative(y(x), x) + (9*x**2 - 9)*y(x) + (2*x**3 + 6*x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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