54.6.9 problem 9

Internal problem ID [8641]
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 : 9
Date solved : Wednesday, March 05, 2025 at 06:11:16 AM
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*} 1 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 38

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=1);
 

\[ y = c_{1} \left (1+\frac {2}{3} \left (x -1\right )+\frac {1}{6} \left (x -1\right )^{2}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right )+\frac {c_{2} \left (-2-8 \left (x -1\right )-12 \left (x -1\right )^{2}-8 \left (x -1\right )^{3}-2 \left (x -1\right )^{4}+\operatorname {O}\left (\left (x -1\right )^{8}\right )\right )}{\left (x -1\right )^{2}} \]

✓ Mathematica. Time used: 0.413 (sec). Leaf size: 52

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,1,7}]
 

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

✓ Sympy. Time used: 0.910 (sec). Leaf size: 14

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=1,n=8)
 

\[ y{\left (x \right )} = \frac {C_{2}}{\left (x - 1\right )^{2}} + C_{1} + O\left (x^{8}\right ) \]