46.3.4 problem 6

Internal problem ID [7325]
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 : 6
Date solved : Sunday, March 30, 2025 at 11:54:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\frac {\left (x +\frac {3}{4}\right ) y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 47

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+1/4*(x+3/4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{{3}/{4}} \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 90

ode=x^2*D[y[x],{x,2}]+1/4*(x+3/4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \sqrt [4]{x} \left (-\frac {x^5}{3628800}+\frac {x^4}{40320}-\frac {x^3}{720}+\frac {x^2}{24}-\frac {x}{2}+1\right )+c_1 x^{3/4} \left (-\frac {x^5}{39916800}+\frac {x^4}{362880}-\frac {x^3}{5040}+\frac {x^2}{120}-\frac {x}{6}+1\right ) \]

✓ Sympy. Time used: 1.062 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + (x + 3/4)*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{4}} \left (\frac {x^{4}}{362880} - \frac {x^{3}}{5040} + \frac {x^{2}}{120} - \frac {x}{6} + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {x^{4}}{40320} - \frac {x^{3}}{720} + \frac {x^{2}}{24} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]