10.13.13 problem 16

Internal problem ID [1373]
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 : 16
Date solved : Saturday, March 29, 2025 at 10:53:36 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*}

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

\[ y = -1+3 x +x^{2}-\frac {3}{4} x^{3}-\frac {1}{6} x^{4}+\frac {21}{160} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to \frac {21 x^5}{160}-\frac {x^4}{6}-\frac {3 x^3}{4}+x^2+3 x-1 \]

✓ Sympy. Time used: 0.778 (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 = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

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