14.28.6 problem 11

Internal problem ID [2798]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.1 (Introduction). Page 377
Problem number : 11
Date solved : Monday, January 27, 2025 at 06:13:56 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )+y+z \left (t \right )-2 \,{\mathrm e}^{-t}\\ y^{\prime }&=2 x \left (t \right )+y-z \left (t \right )-2 \,{\mathrm e}^{-t}\\ z^{\prime }\left (t \right )&=-3 x \left (t \right )+2 y+4 z \left (t \right )+3 \,{\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 0\\ z \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 16

dsolve([diff(x(t),t) = x(t)+y(t)+z(t)-2*exp(-t), diff(y(t),t) = 2*x(t)+y(t)-z(t)-2*exp(-t), diff(z(t),t) = -3*x(t)+2*y(t)+4*z(t)+3*exp(-t), x(0) = 1, y(0) = 0, z(0) = 0], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \\ y &= 0 \\ z \left (t \right ) &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.176 (sec). Leaf size: 18

DSolve[{D[x[t],t]==x[t]+y[t]+z[t]-2*Exp[-t],D[y[t],t]==2*x[t]+y[t]-z[t]-2*Exp[-t],D[z[t],t]==-3*x[t]+2*y[t]+4*z[t]+3*Exp[-t]},{x[0]==1,y[0]==0,z[0]==0},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} \\ y(t)\to 0 \\ z(t)\to 0 \\ \end{align*}