20.28.3 problem 3

Internal problem ID [4065]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Additional problems. Section 11.7. page 788
Problem number : 3
Date solved : Sunday, March 30, 2025 at 02:16:07 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-6 x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

Order:=6; 
ode:=(-x^2+1)*diff(diff(y(x),x),x)-6*x*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (3 x^{4}+2 x^{2}+1\right ) y \left (0\right )+\left (x +\frac {5}{3} x^{3}+\frac {7}{3} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 38

ode=(1-x^2)*D[y[x],{x,2}]-6*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {7 x^5}{3}+\frac {5 x^3}{3}+x\right )+c_1 \left (3 x^4+2 x^2+1\right ) \]

✓ Sympy. Time used: 0.891 (sec). Leaf size: 31

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

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