9.8.8 problem problem 8

Internal problem ID [1073]
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 8
Date solved : Tuesday, March 04, 2025 at 12:08:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} \left (-x^{2}+2\right ) y^{\prime \prime }-x y^{\prime }+16 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 39

Order:=6; 
ode:=(-x^2+2)*diff(diff(y(x),x),x)-x*diff(y(x),x)+16*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (2 x^{4}-4 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {5}{4} x^{3}+\frac {7}{32} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 38

ode=(2-x^2)*D[y[x],{x,2}]-x*D[y[x],x]+16*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {7 x^5}{32}-\frac {5 x^3}{4}+x\right )+c_1 \left (2 x^4-4 x^2+1\right ) \]

✓ Sympy. Time used: 0.848 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (2 - x**2)*Derivative(y(x), (x, 2)) + 16*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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