68.16.2 problem 2

Internal problem ID [17795]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 2
Date solved : Thursday, October 02, 2025 at 02:28:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 69

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

\[ y = \left (1-\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {\left (x +2\right )^{4}}{768}\right ) y \left (-2\right )+\left (x +2+\frac {\left (x +2\right )^{2}}{8}-\frac {\left (x +2\right )^{3}}{48}-\frac {5 \left (x +2\right )^{4}}{768}-\frac {\left (x +2\right )^{5}}{960}\right ) y^{\prime }\left (-2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 78

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

\[ y(x)\to c_1 \left (-\frac {1}{768} (x+2)^4-\frac {1}{48} (x+2)^3-\frac {1}{8} (x+2)^2+1\right )+c_2 \left (-\frac {1}{960} (x+2)^5-\frac {5}{768} (x+2)^4-\frac {1}{48} (x+2)^3+\frac {1}{8} (x+2)^2+x+2\right ) \]

✓ Sympy. Time used: 0.258 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = C_{2} \left (x - \frac {5 \left (x + 2\right )^{4}}{768} - \frac {\left (x + 2\right )^{3}}{48} + \frac {\left (x + 2\right )^{2}}{8} + 2\right ) + C_{1} \left (- \frac {\left (x + 2\right )^{4}}{768} - \frac {\left (x + 2\right )^{3}}{48} - \frac {\left (x + 2\right )^{2}}{8} + 1\right ) + O\left (x^{6}\right ) \]