Internal problem ID [4905]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\frac {\left (x^{2}-1\right ) y}{4}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 35
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);
\[ y \relax (x ) = \frac {c_{1} x \left (1-\frac {1}{24} x^{2}+\frac {1}{1920} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{384} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 58
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+1/4*(x^2-1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^{7/2}}{384}-\frac {x^{3/2}}{8}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{1920}-\frac {x^{5/2}}{24}+\sqrt {x}\right ) \]