40.4.3 problem 3

Internal problem ID [8616]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:39:51 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+9 x y^{\prime }+\left (36 x^{4}-16\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 32

Order:=6; 
ode:=9*x^2*diff(diff(y(x),x),x)+9*x*diff(y(x),x)+(36*x^4-16)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{8}/{3}} \left (1-\frac {3}{20} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {3}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{4}/{3}}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 38

ode=9*x^2*D[y[x],{x,2}]+9*x*D[y[x],x]+(36*x^4-16)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {3 x^4}{20}\right ) x^{4/3}+\frac {c_2 \left (1-\frac {3 x^4}{4}\right )}{x^{4/3}} \]

✓ Sympy. Time used: 0.323 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(9*x**2*Derivative(y(x), (x, 2)) + 9*x*Derivative(y(x), x) + (36*x**4 - 16)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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