46.4.3 problem 3

Internal problem ID [7330]
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 : Monday, January 27, 2025 at 02:50:31 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*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 32

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

\[ y = \frac {c_{2} x^{{8}/{3}} \left (1-\frac {3}{20} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {3}{4} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{4}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 38

AsymptoticDSolveValue[9*x^2*D[y[x],{x,2}]+9*x*D[y[x],x]+(36*x^4-16)*y[x]==0,y[x],{x,0,"6"-1}]
 

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