54.6.8 problem 8

Internal problem ID [8640]
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 : 8
Date solved : Sunday, March 30, 2025 at 01:22:02 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 50

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

\[ y = c_1 \,x^{4} \left (1+2 x +3 x^{2}+4 x^{3}+5 x^{4}+6 x^{5}+7 x^{6}+8 x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (-144-96 x -48 x^{2}+48 x^{4}+96 x^{5}+144 x^{6}+192 x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.388 (sec). Leaf size: 77

ode=x*(1-x)*D[y[x],{x,2}]-3*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-x^6-\frac {2 x^5}{3}-\frac {x^4}{3}+\frac {x^2}{3}+\frac {2 x}{3}+1\right )+c_2 \left (7 x^{10}+6 x^9+5 x^8+4 x^7+3 x^6+2 x^5+x^4\right ) \]

✓ Sympy. Time used: 0.923 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} x^{4} \left (\frac {2 x^{3}}{315} + \frac {x^{2}}{15} + \frac {2 x}{5} + 1\right ) + O\left (x^{8}\right ) \]