78.3.20 problem 5.d

Internal problem ID [21016]
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.d
Date solved : Thursday, October 02, 2025 at 07:01:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 90

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

\[ y(x)\to c_2 e^{4/x} \left (-\frac {158133 x^5}{1310720}-\frac {2701 x^4}{32768}-\frac {61 x^3}{1024}-\frac {3 x^2}{128}+\frac {x}{8}+1\right ) x^2+c_1 \left (\frac {6993 x^5}{1310720}+\frac {483 x^4}{32768}+\frac {49 x^3}{1024}+\frac {21 x^2}{128}+\frac {3 x}{8}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x - 2)*Derivative(y(x), (x, 2)) + (4*x - 8)*Derivative(y(x), x) + 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

ValueError : ODE x**2*(x - 2)*Derivative(y(x), (x, 2)) + (4*x - 8)*Derivative(y(x), x) + 3*y(x) does not match hint 2nd_power_series_regular