6.13.29 problem 31(c)

Internal problem ID [1920]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 31(c)
Date solved : Tuesday, September 30, 2025 at 05:21:18 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (4 x^{2}-4 x +1\right ) y^{\prime \prime }-\left (8-16 x \right ) y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 59

Order:=6; 
ode:=(4*x^2-4*x+1)*diff(diff(y(x),x),x)-(8-16*x)*diff(y(x),x)+8*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (-128 x^{5}-48 x^{4}-16 x^{3}-4 x^{2}+1\right ) y \left (0\right )+\left (80 x^{5}+32 x^{4}+12 x^{3}+4 x^{2}+x \right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 54

ode=(1-4*x+4*x^2)*D[y[x],{x,2}]-(8-16*x)*D[y[x],x]+8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-128 x^5-48 x^4-16 x^3-4 x^2+1\right )+c_2 \left (80 x^5+32 x^4+12 x^3+4 x^2+x\right ) \]

✓ Sympy. Time used: 0.281 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((16*x - 8)*Derivative(y(x), x) + (4*x**2 - 4*x + 1)*Derivative(y(x), (x, 2)) + 8*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- 48 x^{4} - 16 x^{3} - 4 x^{2} + 1\right ) + C_{1} x \left (32 x^{3} + 12 x^{2} + 4 x + 1\right ) + O\left (x^{6}\right ) \]