23.5.3 problem 3(b)

Internal problem ID [4180]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(b)
Date solved : Sunday, March 30, 2025 at 02:41:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 44

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

\[ y = c_1 \,x^{2} \left (1-\frac {1}{3} x +\frac {1}{12} x^{2}-\frac {1}{60} x^{3}+\frac {1}{360} x^{4}-\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 69

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

\[ \{y(x)\}\to c_1 \left (\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right )+c_2 \left (\frac {x^6}{360}-\frac {x^5}{60}+\frac {x^4}{12}-\frac {x^3}{3}+x^2\right ) \]

✓ Sympy. Time used: 0.883 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{2} \left (1 - x\right ) + C_{1} x^{2} \left (- \frac {x^{3}}{60} + \frac {x^{2}}{12} - \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]