78.3.22 problem 6.c.ii

Internal problem ID [21018]
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 : 6.c.ii
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*} \infty \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 122

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=infinity);
 

\[ y = \frac {28800 x \left (O\left (\frac {1}{x^{6}}\right ) x^{5}-7 x^{4}+\frac {21 x^{3}}{2}+\frac {35 x^{2}}{4}+\frac {259 x}{48}+\frac {609}{320}\right ) c_2 \ln \left (\frac {1}{x}\right )+28800 x^{5} \left (c_2 x +c_1 \right ) O\left (\frac {1}{x^{6}}\right )+28800 c_2 \,x^{6}+\left (28800 c_1 +115200 c_2 \right ) x^{5}+\left (-43200 c_1 -1116000 c_2 \right ) x^{4}+\left (-36000 c_1 -367200 c_2 \right ) x^{3}+\left (-22200 c_1 +225550 c_2 \right ) x^{2}+\left (-7830 c_1 +680313 c_2 \right ) x +3927 c_1}{28800 x^{5}} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 93

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,Infinity,5}]
 

\[ y(x)\to c_1 \left (\frac {2291}{576 x^3}-\frac {259 \log (x)}{48 x^3}-\frac {19}{x^2}-\frac {35 \log (x)}{4 x^2}+x-\frac {185}{4 x}+7 \log (x)-\frac {21 \log (x)}{2 x}+9\right )+c_2 \left (-\frac {87}{320 x^4}-\frac {37}{48 x^3}-\frac {5}{4 x^2}-\frac {3}{2 x}+1\right ) \]

✓ Sympy. Time used: 6.463 (sec). Leaf size: 529

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_ordinary",x0=inf,n=6)
 

\[ \text {Solution too large to show} \]