40.3.3 problem 4

Internal problem ID [8610]
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 : 4
Date solved : Tuesday, September 30, 2025 at 05:39:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left ({\mathrm e}^{-2 x}-\frac {1}{9}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 54

Order:=6; 
ode:=diff(diff(y(x),x),x)+(exp(-2*x)-1/9)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {4}{9} x^{2}+\frac {1}{3} x^{3}-\frac {65}{486} x^{4}+\frac {1}{135} x^{5}\right ) y \left (0\right )+\left (x -\frac {4}{27} x^{3}+\frac {1}{6} x^{4}-\frac {227}{2430} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 63

ode=D[y[x],{x,2}]+(Exp[-2*x]-1/9)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {227 x^5}{2430}+\frac {x^4}{6}-\frac {4 x^3}{27}+x\right )+c_1 \left (\frac {x^5}{135}-\frac {65 x^4}{486}+\frac {x^3}{3}-\frac {4 x^2}{9}+1\right ) \]

✓ Sympy. Time used: 0.335 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-1/9 + exp(-2*x))*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} \left (e^{2 x} - 9\right )^{2} e^{- 4 x}}{1944} + \frac {x^{2}}{18} - \frac {x^{2} e^{- 2 x}}{2} + 1\right ) + C_{1} x \left (\frac {x^{2}}{54} - \frac {x^{2} e^{- 2 x}}{6} + 1\right ) + O\left (x^{6}\right ) \]