46.4.2 problem 2

Internal problem ID [7329]
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 5.5. Bessel Functions Y(x). General Solution page 200
Problem number : 2
Date solved : Sunday, March 30, 2025 at 11:54:31 AM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+5 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+5*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x^{4}+c_2 \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{4}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 47

ode=x*D[y[x],{x,2}]+5*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{384}-\frac {x^2}{12}+1\right )+c_1 \left (\frac {\left (x^2+8\right )^2}{64 x^4}-\frac {\log (x)}{16}\right ) \]

✓ Sympy. Time used: 0.856 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + x*Derivative(y(x), (x, 2)) + 5*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{4}}{384} - \frac {x^{2}}{12} + 1\right ) + O\left (x^{6}\right ) \]