9.8.18 problem problem 18

Internal problem ID [1083]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.2 Power series solutions. Page 624
Problem number : problem 18
Date solved : Saturday, March 29, 2025 at 10:38:28 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 14

Order:=6; 
ode:=diff(diff(y(x),x),x)+(x-1)*diff(y(x),x)+y(x) = 0; 
ic:=y(1) = 2, D(y)(1) = 0; 
dsolve([ode,ic],y(x),type='series',x=1);
 

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

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 21

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

\[ y(x)\to \frac {1}{4} (x-1)^4-(x-1)^2+2 \]

✓ Sympy. Time used: 0.718 (sec). Leaf size: 34

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

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