65.13.5 problem 20.2 (iv) (k=-2)

Internal problem ID [13732]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 20, Series solutions of second order linear equations. Exercises page 195
Problem number : 20.2 (iv) (k=-2)
Date solved : Wednesday, March 05, 2025 at 10:14:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-2*x*D[y[x],x]-4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.677 (sec). Leaf size: 29

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

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