46.4.6 problem 6

Internal problem ID [7333]
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 : 6
Date solved : Wednesday, March 05, 2025 at 04:23:33 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+36 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+diff(y(x),x)+36*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-36 x +324 x^{2}-1296 x^{3}+2916 x^{4}-\frac {104976}{25} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (72 x -972 x^{2}+4752 x^{3}-12150 x^{4}+\frac {2396952}{125} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 93

ode=x*D[y[x],{x,2}]+D[y[x],x]+36*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {104976 x^5}{25}+2916 x^4-1296 x^3+324 x^2-36 x+1\right )+c_2 \left (\frac {2396952 x^5}{125}-12150 x^4+4752 x^3-972 x^2+\left (-\frac {104976 x^5}{25}+2916 x^4-1296 x^3+324 x^2-36 x+1\right ) \log (x)+72 x\right ) \]

✓ Sympy. Time used: 0.815 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 36*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {104976 x^{5}}{25} + 2916 x^{4} - 1296 x^{3} + 324 x^{2} - 36 x + 1\right ) + O\left (x^{6}\right ) \]