23.5.11 problem 3(j)

Internal problem ID [4188]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(j)
Date solved : Tuesday, March 04, 2025 at 05:55:10 PM
CAS classification : [_Jacobi]

\begin{align*} y^{\prime \prime }+\frac {\left (1-5 x \right ) y^{\prime }}{-x^{2}+x}-\frac {4 y}{-x^{2}+x}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.014 (sec). Leaf size: 44

Order:=6; 
ode:=diff(diff(y(x),x),x)+(1-5*x)/(-x^2+x)*diff(y(x),x)-4/(-x^2+x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+4 x +9 x^{2}+16 x^{3}+25 x^{4}+36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -12 x^{2}-24 x^{3}-40 x^{4}-60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 88

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

\[ \{y(x)\}\to c_1 \left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right )+c_2 \left (-60 x^5-40 x^4-24 x^3-12 x^2+\left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right ) \log (x)-4 x\right ) \]

✓ Sympy. Time used: 0.970 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - 5*x)*Derivative(y(x), x)/(-x**2 + x) + Derivative(y(x), (x, 2)) - 4*y(x)/(-x**2 + x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {16 x^{5}}{225} + \frac {4 x^{4}}{9} - \frac {16 x^{3}}{9} + 4 x^{2} - 4 x + 1\right ) + O\left (x^{6}\right ) \]