9.7.22 problem problem 22

Internal problem ID [1063]
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 22
Date solved : Saturday, March 29, 2025 at 10:38:00 PM
CAS classification : [[_2nd_order, _missing_x]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = 1-2 x +2 x^{2}-\frac {4}{3} x^{3}+\frac {2}{3} x^{4}-\frac {4}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {4 x^5}{15}+\frac {2 x^4}{3}-\frac {4 x^3}{3}+2 x^2-2 x+1 \]

✓ Sympy. Time used: 0.735 (sec). Leaf size: 42

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

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