12.16.2 problem Example 7.7.2 page 383

Internal problem ID [2064]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : Example 7.7.2 page 383
Date solved : Saturday, March 29, 2025 at 11:47:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }+x \left (8-9 x \right ) y^{\prime }+\left (6-3 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: 45

Order:=6; 
ode:=x^2*(1-2*x)*diff(diff(y(x),x),x)+x*(8-9*x)*diff(y(x),x)+(6-3*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{3} x -\frac {1}{14} x^{2}-\frac {1}{28} x^{3}-\frac {25}{1008} x^{4}-\frac {1}{48} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_2 \left (2880-23760 x +71280 x^{2}-83160 x^{3}+62370 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{6}} \]

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 61

ode=x^2*(1-2*x)*D[y[x],{x,2}]+x*(8-9*x)*D[y[x],x]+(6-3*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {25 x^3}{1008}-\frac {x^2}{28}-\frac {x}{14}+\frac {1}{x}-\frac {1}{3}\right )+c_1 \left (\frac {1}{x^6}-\frac {33}{4 x^5}+\frac {99}{4 x^4}-\frac {231}{8 x^3}\right ) \]

✓ Sympy. Time used: 1.127 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x)*Derivative(y(x), (x, 2)) + x*(8 - 9*x)*Derivative(y(x), x) + (6 - 3*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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