54.3.28 problem 28

Internal problem ID [8584]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 28
Date solved : Wednesday, March 05, 2025 at 06:10:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

\[ y = \frac {\left (-x^{3}+3 x^{2}-2\right ) y^{\prime }\left (1\right )}{3}-3 \left (x^{2}-2 x +\frac {2}{3}\right ) y \left (1\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 31

ode=(x^2-2*x+2)*D[y[x],{x,2}]-4*(x-1)*D[y[x],x]+6*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,7}]
 

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

✓ Sympy. Time used: 0.830 (sec). Leaf size: 27

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

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