54.6.14 problem 14

Internal problem ID [8646]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number : 14
Date solved : Wednesday, March 05, 2025 at 06:11:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 48

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

\[ y = c_{1} x^{5} \left (1+7 x +32 x^{2}+120 x^{3}+400 x^{4}+1232 x^{5}+3584 x^{6}+9984 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (2880+5760 x +5760 x^{2}+92160 x^{5}+645120 x^{6}+2949120 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 64

ode=x*(1-2*x)*D[y[x],{x,2}]-2*(2+x)*D[y[x],x]+8*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (224 x^6+32 x^5+2 x^2+2 x+1\right )+c_2 \left (3584 x^{11}+1232 x^{10}+400 x^9+120 x^8+32 x^7+7 x^6+x^5\right ) \]

✓ Sympy. Time used: 1.005 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = C_{1} x^{5} \left (\frac {16 x^{2}}{21} + \frac {4 x}{3} + 1\right ) + O\left (x^{8}\right ) \]