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 : Wednesday, March 05, 2025 at 04:23:26 AM
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 (x +1\right ) y&=0 \end{align*}

Using series method with expansion around

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

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

Order:=6; 
ode:=(2*x+1)^2*diff(diff(y(x),x),x)+2*(2*x+1)*diff(y(x),x)+16*x*(1+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=(2*x+1)^2*D[y[x],{x,2}]+2*(2*x+1)*D[y[x],x]+16*x*(x+1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 1.084 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(16*x*(x + 1)*y(x) + (2*x + 1)**2*Derivative(y(x), (x, 2)) + (4*x + 2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = - \frac {5 x^{4} r{\left (3 \right )}}{2} + \frac {26 x^{5} r{\left (3 \right )}}{5} + C_{2} \left (- 8 x^{5} + 4 x^{4} - 4 x^{2} + 1\right ) + C_{1} x \left (1 - x\right ) + O\left (x^{6}\right ) \]