12.12.19 problem 21

Internal problem ID [1873]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 21
Date solved : Tuesday, March 04, 2025 at 01:45:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}-4\right ) y^{\prime \prime }-x y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

Order:=6; 
ode:=(x^2-4)*diff(diff(y(x),x),x)-x*diff(y(x),x)-3*y(x) = 0; 
ic:=y(0) = -1, D(y)(0) = 2; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = -1+2 x +\frac {3}{8} x^{2}-\frac {1}{3} x^{3}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

ode=(x^2-4)*D[y[x],{x,2}]-x*D[y[x],x]-3*y[x]==0; 
ic={y[0]==-1,Derivative[1][y][0] ==2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to -\frac {3 x^4}{128}-\frac {x^3}{3}+\frac {3 x^2}{8}+2 x-1 \]

✓ Sympy. Time used: 0.812 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (x**2 - 4)*Derivative(y(x), (x, 2)) - 3*y(x),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {3 x^{4}}{128} - \frac {3 x^{2}}{8} + 1\right ) + C_{1} x \left (1 - \frac {x^{2}}{6}\right ) + O\left (x^{6}\right ) \]