73.23.22 problem 33.5 (j)

Internal problem ID [15597]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.5 (j)
Date solved : Monday, March 31, 2025 at 01:42:01 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 41

Order:=6; 
ode:=(x^2-2*x+2)*diff(diff(y(x),x),x)+(1-x)*diff(y(x),x)-3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=1);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode=(x^2-2*x+2)*D[y[x],{x,2}]+(1-x)*D[y[x],x]-3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,1,5}]
 

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

✓ Sympy. Time used: 0.926 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - x)*Derivative(y(x), x) + (x**2 - 2*x + 2)*Derivative(y(x), (x, 2)) - 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=1,n=6)
 

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