46.3.1 problem 2

Internal problem ID [7322]
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 5.4. Bessels Equation page 195
Problem number : 2
Date solved : Monday, January 27, 2025 at 02:50:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {4}{49}\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: 36

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-4/49)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} x^{{4}/{7}} \left (1-\frac {7}{36} x^{2}+\frac {49}{4608} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {7}{20} x^{2}+\frac {49}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{7}}} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 52

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-4/49)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 x^{2/7} \left (\frac {49 x^4}{4608}-\frac {7 x^2}{36}+1\right )+\frac {c_2 \left (\frac {49 x^4}{1920}-\frac {7 x^2}{20}+1\right )}{x^{2/7}} \]