Internal problem ID [4907]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Bessel]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-6\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 97
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-6)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\sqrt {6}} \left (1+\frac {1}{-4+4 \sqrt {6}} x^{2}+\frac {1}{32} \frac {1}{\left (-2+\sqrt {6}\right ) \left (-1+\sqrt {6}\right )} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {6}} \left (1-\frac {1}{4+4 \sqrt {6}} x^{2}+\frac {1}{32} \frac {1}{\left (2+\sqrt {6}\right ) \left (1+\sqrt {6}\right )} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 210
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-6)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^4}{\left (-4-\sqrt {6}+\left (1-\sqrt {6}\right ) \left (2-\sqrt {6}\right )\right ) \left (-2-\sqrt {6}+\left (3-\sqrt {6}\right ) \left (4-\sqrt {6}\right )\right )}-\frac {x^2}{-4-\sqrt {6}+\left (1-\sqrt {6}\right ) \left (2-\sqrt {6}\right )}+1\right ) x^{-\sqrt {6}}+c_1 \left (\frac {x^4}{\left (-4+\sqrt {6}+\left (1+\sqrt {6}\right ) \left (2+\sqrt {6}\right )\right ) \left (-2+\sqrt {6}+\left (3+\sqrt {6}\right ) \left (4+\sqrt {6}\right )\right )}-\frac {x^2}{-4+\sqrt {6}+\left (1+\sqrt {6}\right ) \left (2+\sqrt {6}\right )}+1\right ) x^{\sqrt {6}} \]