Internal problem ID [4906]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (2 x +1\right )^{2} y^{\prime \prime }+2 \left (2 x +1\right ) y^{\prime }+16 x \left (1+x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 49
Order:=6; dsolve((2*x+1)^2*diff(y(x),x$2)+2*(2*x+1)*diff(y(x),x)+16*x*(x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {8}{3} x^{3}+\frac {16}{3} x^{4}-\frac {152}{15} x^{5}\right ) y \relax (0)+\left (x -x^{2}+\frac {4}{3} x^{3}-\frac {10}{3} x^{4}+\frac {104}{15} x^{5}\right ) D\relax (y )\relax (0)+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 61
AsymptoticDSolveValue[(2*x+1)^2*y''[x]+2*(2*x+1)*y'[x]+16*x*(x+1)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {152 x^5}{15}+\frac {16 x^4}{3}-\frac {8 x^3}{3}+1\right )+c_2 \left (\frac {104 x^5}{15}-\frac {10 x^4}{3}+\frac {4 x^3}{3}-x^2+x\right ) \]