46.3.6 problem 8

Internal problem ID [7327]
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
Date solved : Monday, January 27, 2025 at 02:50:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \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{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 54

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);
 

\[ y = \left (1-\frac {8}{3} x^{3}+\frac {16}{3} x^{4}-\frac {152}{15} x^{5}\right ) y \left (0\right )+\left (x -x^{2}+\frac {4}{3} x^{3}-\frac {10}{3} x^{4}+\frac {104}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 61

AsymptoticDSolveValue[(2*x+1)^2*D[y[x],{x,2}]+2*(2*x+1)*D[y[x],x]+16*x*(x+1)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {152 x^5}{15}+\frac {16 x^4}{3}-\frac {8 x^3}{3}+1\right )+c_2 \left (\frac {104 x^5}{15}-\frac {10 x^4}{3}+\frac {4 x^3}{3}-x^2+x\right ) \]