14.26.9 problem 1

Internal problem ID [4034]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page 771
Problem number : 1
Date solved : Tuesday, September 30, 2025 at 07:00:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 48

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

\[ y = x^{2} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {11}{36} x^{3}-\frac {25}{288} x^{4}+\frac {137}{7200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 120

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

\[ y(x)\to c_1 \left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) x^2+c_2 \left (\left (\frac {137 x^5}{7200}-\frac {25 x^4}{288}+\frac {11 x^3}{36}-\frac {3 x^2}{4}+x\right ) x^2+\left (-\frac {x^5}{120}+\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) x^2 \log (x)\right ) \]

✓ Sympy. Time used: 0.324 (sec). Leaf size: 24

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

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