10.7.19 problem 21

Internal problem ID [1267]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.1 Homogeneous Equations with Constant Coefficients, page 144
Problem number : 21
Date solved : Saturday, March 29, 2025 at 10:50:42 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-diff(y(x),x)-2*y(x) = 0; 
ic:=y(0) = alpha, D(y)(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (-2+2 \alpha \right ) {\mathrm e}^{-x}}{3}+\frac {\left (\alpha +2\right ) {\mathrm e}^{2 x}}{3} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 29

ode=D[y[x],{x,2}]-D[y[x],x]-2*y[x]==0; 
ic={y[0]==\[Alpha],Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{3} e^{-x} \left (2 (\alpha -1)+(\alpha +2) e^{3 x}\right ) \]

✓ Sympy. Time used: 0.165 (sec). Leaf size: 26

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): alpha, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {\alpha }{3} + \frac {2}{3}\right ) e^{2 x} + \left (\frac {2 \alpha }{3} - \frac {2}{3}\right ) e^{- x} \]