23.5.5 problem 3(d)

Internal problem ID [4182]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(d)
Date solved : Sunday, March 30, 2025 at 02:41:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+\left (\frac {1}{4 x^{2}}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 46

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

\[ y = \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x +x^{2}+\frac {2}{3} x^{3}+\frac {17}{48} x^{4}+\frac {37}{240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {1}{2} x^{2}-\frac {1}{2} x^{3}-\frac {11}{32} x^{4}-\frac {17}{96} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 114

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

\[ \{y(x)\}\to c_1 \sqrt {x} \left (\frac {37 x^5}{240}+\frac {17 x^4}{48}+\frac {2 x^3}{3}+x^2+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {17 x^5}{96}-\frac {11 x^4}{32}-\frac {x^3}{2}-\frac {x^2}{2}\right )+\sqrt {x} \left (\frac {37 x^5}{240}+\frac {17 x^4}{48}+\frac {2 x^3}{3}+x^2+x+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.881 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-1 + 1/(4*x**2))*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} \left (\frac {17 x^{4}}{48} + \frac {2 x^{3}}{3} + x^{2} + x + 1\right ) + O\left (x^{6}\right ) \]