46.4.7 problem 7

Internal problem ID [7334]
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 : 7
Date solved : Monday, January 27, 2025 at 02:50:35 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*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 35

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

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[D[y[x],{x,2}]+k^2*x^2*y[x]==0,y[x],{x,0,"6"-1}]
 

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