40.3.2 problem 3

Internal problem ID [8609]
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 : 3
Date solved : Tuesday, September 30, 2025 at 05:39:45 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+y^{\prime }+\frac {y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 44

Order:=6; 
ode:=x*diff(diff(y(x),x),x)+diff(y(x),x)+1/4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{4} x +\frac {1}{64} x^{2}-\frac {1}{2304} x^{3}+\frac {1}{147456} x^{4}-\frac {1}{14745600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x -\frac {3}{64} x^{2}+\frac {11}{6912} x^{3}-\frac {25}{884736} x^{4}+\frac {137}{442368000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 117

ode=x*D[y[x],{x,2}]+D[y[x],x]+1/4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right )+c_2 \left (\frac {137 x^5}{442368000}-\frac {25 x^4}{884736}+\frac {11 x^3}{6912}-\frac {3 x^2}{64}+\left (-\frac {x^5}{14745600}+\frac {x^4}{147456}-\frac {x^3}{2304}+\frac {x^2}{64}-\frac {x}{4}+1\right ) \log (x)+\frac {x}{2}\right ) \]

✓ Sympy. Time used: 0.200 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{5}}{14745600} + \frac {x^{4}}{147456} - \frac {x^{3}}{2304} + \frac {x^{2}}{64} - \frac {x}{4} + 1\right ) + O\left (x^{6}\right ) \]