84.31.3 problem 19.3

Internal problem ID [22301]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 19. Power series solutions about an ordinary point. Solved problems. Page 98
Problem number : 19.3
Date solved : Thursday, October 02, 2025 at 08:37:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

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

\[ y = \left (1-\left (x -2\right )^{2}\right ) y \left (2\right )+\left (x -2-\frac {\left (x -2\right )^{3}}{6}-\frac {\left (x -2\right )^{5}}{120}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (1-(x-2)^2\right )+c_2 \left (-\frac {1}{120} (x-2)^5-\frac {1}{6} (x-2)^3+x-2\right ) \]

✓ Sympy. Time used: 0.187 (sec). Leaf size: 26

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

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