23.5.7 problem 3(f)

Internal problem ID [4184]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(f)
Date solved : Sunday, March 30, 2025 at 02:41:52 AM
CAS classification : [_Jacobi, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\frac {3 y^{\prime }}{x \left (1-x \right )}+\frac {2 y}{x \left (1-x \right )}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 42

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

\[ y = c_1 \,x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-144-96 x -48 x^{2}+48 x^{4}+96 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 56

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

\[ \{y(x)\}\to c_1 \left (-\frac {x^4}{3}+\frac {x^2}{3}+\frac {2 x}{3}+1\right )+c_2 \left (5 x^8+4 x^7+3 x^6+2 x^5+x^4\right ) \]

✓ Sympy. Time used: 0.884 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + 2*y(x)/(x*(1 - x)) - 3*Derivative(y(x), x)/(x*(1 - x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{4} \left (\frac {2 x}{5} + 1\right ) + O\left (x^{6}\right ) \]