46.3.5 problem 7

Internal problem ID [7326]
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 : 7
Date solved : Monday, January 27, 2025 at 02:50:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 34

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+1/4*(x^2-1)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x \left (1-\frac {1}{24} x^{2}+\frac {1}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 58

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

\[ y(x)\to c_1 \left (\frac {x^{7/2}}{384}-\frac {x^{3/2}}{8}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{1920}-\frac {x^{5/2}}{24}+\sqrt {x}\right ) \]