67.25.1 problem 35.2 (a)

Internal problem ID [16996]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 35. Modified Power series solutions and basic method of Frobenius. Additional Exercises. page 715
Problem number : 35.2 (a)
Date solved : Thursday, October 02, 2025 at 01:41:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 24

Order:=6; 
ode:=(x-3)^2*diff(diff(y(x),x),x)-2*(x-3)*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (-x^{2}+9\right ) y \left (0\right )}{9}-\frac {y^{\prime }\left (0\right ) x \left (x -3\right )}{3} \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=(x-3)^2*D[y[x],{x,2}]-2*(x-3)*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (1-\frac {x^2}{9}\right )+c_2 \left (x-\frac {x^2}{3}\right ) \]

✓ Sympy. Time used: 0.253 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 3)**2*Derivative(y(x), (x, 2)) - (2*x - 6)*Derivative(y(x), x) + 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (1 - \frac {x^{2}}{9}\right ) + C_{1} x \left (1 - \frac {x}{3}\right ) + O\left (x^{6}\right ) \]