10.13.5 problem 6

Internal problem ID [1365]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.2, Series Solutions Near an Ordinary Point, Part I. page 263
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 12:34:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}+2\right ) y^{\prime \prime }-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: 39

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

\[ y = \left (1-x^{2}+\frac {1}{6} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{3}+\frac {7}{160} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_2 \left (\frac {7 x^5}{160}-\frac {x^3}{4}+x\right )+c_1 \left (\frac {x^4}{6}-x^2+1\right ) \]

✓ Sympy. Time used: 0.804 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + (x**2 + 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)
 

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