10.13.3 problem 4

Internal problem ID [1363]
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 : 4
Date solved : Saturday, March 29, 2025 at 10:53:23 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+k^{2} x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 35

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

\[ y = \left (1-\frac {k^{2} x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} k^{2} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+k^2*x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (x-\frac {k^2 x^5}{20}\right )+c_1 \left (1-\frac {k^2 x^4}{12}\right ) \]

✓ Sympy. Time used: 0.745 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
k = symbols("k") 
y = Function("y") 
ode = Eq(k**2*x**2*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)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {k^{2} x^{4}}{12} + 1\right ) + C_{1} x \left (- \frac {k^{2} x^{4}}{20} + 1\right ) + O\left (x^{6}\right ) \]