78.3.18 problem 5.b

Internal problem ID [21014]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 5.b
Date solved : Thursday, October 02, 2025 at 07:01:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \frac {y^{\prime \prime }}{x}+\frac {3 \left (x -4\right ) y^{\prime }}{6+x}+\frac {x^{2} \left (x -2\right ) y}{x -1}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.081 (sec). Leaf size: 39

Order:=6; 
ode:=1/x*diff(diff(y(x),x),x)+3*(x-4)/(x+6)*diff(y(x),x)+x^2*(x-2)/(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{5}}{10}\right ) y \left (0\right )+\left (x +\frac {1}{3} x^{3}-\frac {5}{72} x^{4}+\frac {77}{720} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode=1/x*D[y[x],{x,2}]+3*(x-4)/(x+6)*D[y[x],x]+x^2*(x-2)/(x-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {x^5}{10}\right )+c_2 \left (\frac {77 x^5}{720}-\frac {5 x^4}{72}+\frac {x^3}{3}+x\right ) \]

✓ Sympy. Time used: 0.491 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x - 2)*y(x)/(x - 1) + (3*x - 12)*Derivative(y(x), x)/(x + 6) + Derivative(y(x), (x, 2))/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {17 x^{8}}{2016} + \frac {5 x^{7}}{252} - \frac {x^{2}}{3} + 1\right ) + C_{1} x \left (\frac {13 x^{7}}{1008} + \frac {x^{6}}{63} + 1\right ) + O\left (x^{6}\right ) \]