23.5.9 problem 3(h)

Internal problem ID [4186]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(h)
Date solved : Tuesday, March 04, 2025 at 05:55:08 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{2 x}+\frac {y}{4 x}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 44

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

\[ y \left (x \right ) = c_{1} x^{{3}/{2}} \left (1-\frac {1}{10} x +\frac {1}{280} x^{2}-\frac {1}{15120} x^{3}+\frac {1}{1330560} x^{4}-\frac {1}{172972800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{2} x -\frac {1}{8} x^{2}+\frac {1}{144} x^{3}-\frac {1}{5760} x^{4}+\frac {1}{403200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 86

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

\[ \{y(x)\}\to c_2 \left (\frac {x^5}{403200}-\frac {x^4}{5760}+\frac {x^3}{144}-\frac {x^2}{8}+\frac {x}{2}+1\right )+c_1 \left (-\frac {x^5}{172972800}+\frac {x^4}{1330560}-\frac {x^3}{15120}+\frac {x^2}{280}-\frac {x}{10}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.842 (sec). Leaf size: 54

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{5}}{403200} - \frac {x^{4}}{5760} + \frac {x^{3}}{144} - \frac {x^{2}}{8} + \frac {x}{2} + 1\right ) + C_{1} x^{\frac {3}{2}} \left (- \frac {x^{3}}{15120} + \frac {x^{2}}{280} - \frac {x}{10} + 1\right ) + O\left (x^{6}\right ) \]