23.5.12 problem 3(k)

Internal problem ID [4189]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(k)
Date solved : Tuesday, March 04, 2025 at 05:55:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 44

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

\[ y \left (x \right ) = c_{1} x^{2} \left (1-x +x^{2}-x^{3}+x^{4}-x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (-2+2 x -2 x^{2}+2 x^{3}-2 x^{4}+2 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 47

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

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

✓ Sympy. Time used: 0.869 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} x^{2} \left (\frac {x^{3}}{360} + \frac {x^{2}}{24} + \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]