9.7.21 problem problem 21

Internal problem ID [1062]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Chapter 11 Power series methods. Section 11.1 Introduction and Review of power series. Page 615
Problem number : problem 21
Date solved : Saturday, March 29, 2025 at 10:37:59 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

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

\[ y = x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 29

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

\[ y(x)\to \frac {x^5}{24}+\frac {x^4}{6}+\frac {x^3}{2}+x^2+x \]

✓ Sympy. Time used: 0.734 (sec). Leaf size: 41

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

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