12.15.24 problem 20

Internal problem ID [2022]
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 II. Exercises 7.6. Page 374
Problem number : 20
Date solved : Saturday, March 29, 2025 at 11:46:04 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (1-4 x \right ) y^{\prime \prime }+3 x \left (1-6 x \right ) y^{\prime }+\left (1-12 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 48

Order:=6; 
ode:=x^2*(1-4*x)*diff(diff(y(x),x),x)+3*x*(1-6*x)*diff(y(x),x)+(1-12*x)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (2 x +7 x^{2}+\frac {74}{3} x^{3}+\frac {533}{6} x^{4}+\frac {1627}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +\left (1+2 x +6 x^{2}+20 x^{3}+70 x^{4}+252 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \left (c_2 \ln \left (x \right )+c_1 \right )}{x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 104

ode=x^2*(1-4*x)*D[y[x],{x,2}]+3*x*(1-6*x)*D[y[x],x]+(1-12*x)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (252 x^5+70 x^4+20 x^3+6 x^2+2 x+1\right )}{x}+c_2 \left (\frac {\frac {1627 x^5}{5}+\frac {533 x^4}{6}+\frac {74 x^3}{3}+7 x^2+2 x}{x}+\frac {\left (252 x^5+70 x^4+20 x^3+6 x^2+2 x+1\right ) \log (x)}{x}\right ) \]

✓ Sympy. Time used: 1.084 (sec). Leaf size: 8

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 4*x)*Derivative(y(x), (x, 2)) + 3*x*(1 - 6*x)*Derivative(y(x), x) + (1 - 12*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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