Internal problem ID [4901]
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.
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 }+\left (x^{2}-\frac {4}{49}\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 35
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);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{7}} \left (1-\frac {7}{36} x^{2}+\frac {49}{4608} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {7}{20} x^{2}+\frac {49}{1920} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {2}{7}}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 52
AsymptoticDSolveValue[x^2*y''[x]+x*y'[x]+(x^2-4/49)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x^{2/7} \left (\frac {49 x^4}{4608}-\frac {7 x^2}{36}+1\right )+\frac {c_2 \left (\frac {49 x^4}{1920}-\frac {7 x^2}{20}+1\right )}{x^{2/7}} \]