46.4.8 problem 8

Internal problem ID [7335]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.5. Bessel Functions Y(x). General Solution page 200
Problem number : 8
Date solved : Sunday, March 30, 2025 at 11:54:40 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 15

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

\[ y = y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 10

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

\[ y(x)\to c_2 x+c_1 \]

✓ Sympy. Time used: 0.805 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {k^{2} x^{6}}{30} + 1\right ) + C_{1} x \left (- \frac {k^{2} x^{6}}{42} + 1\right ) + O\left (x^{6}\right ) \]