73.23.20 problem 33.5 (h)

Internal problem ID [15595]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (h)
Date solved : Monday, March 31, 2025 at 01:41:58 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 44

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

\[ y = c_1 \left (1+\frac {1}{6} x +\frac {1}{36} x^{2}+\frac {1}{216} x^{3}+\frac {1}{1296} x^{4}+\frac {1}{7776} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1+\frac {1}{3} x +\frac {1}{18} x^{2}+\frac {1}{108} x^{3}+\frac {1}{648} x^{4}+\frac {1}{3888} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 64

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

\[ y(x)\to c_1 \left (\frac {x^3}{1296}+\frac {x^2}{216}+\frac {x}{36}+\frac {1}{x}+\frac {1}{6}\right )+c_2 \left (\frac {x^4}{1296}+\frac {x^3}{216}+\frac {x^2}{36}+\frac {x}{6}+1\right ) \]

✓ Sympy. Time used: 0.928 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((4*x - 12)*Derivative(y(x), x) + (x**2 - 6*x)*Derivative(y(x), (x, 2)) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{5}}{2700} + \frac {x^{4}}{180} - \frac {x^{3}}{18} + \frac {x^{2}}{3} - x + 1\right ) + O\left (x^{6}\right ) \]