28.4.25 problem 7.25

Internal problem ID [4557]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.25
Date solved : Monday, January 27, 2025 at 09:23:27 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )-2 x \left (t \right )-y&=2 \,{\mathrm e}^{t}\\ y^{\prime }-2 y-4 z \left (t \right )&=4 \,{\mathrm e}^{2 t}\\ x \left (t \right )-z^{\prime }\left (t \right )-z \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 9\\ y \left (0\right ) = 3\\ z \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 65

dsolve([diff(x(t),t)-2*x(t)-y(t) = 2*exp(t), diff(y(t),t)-2*y(t)-4*z(t) = 4*exp(2*t), x(t)-diff(z(t),t)-z(t) = 0, x(0) = 9, y(0) = 3, z(0) = 1], singsol=all)
 

\begin{align*} x &= 8 \,{\mathrm e}^{3 t}+2 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{2 t}+2+3 t \\ y &= 8 \,{\mathrm e}^{3 t}-4 \,{\mathrm e}^{t}-1-6 t \\ z &= 2 \,{\mathrm e}^{3 t}+{\mathrm e}^{t}-{\mathrm e}^{2 t}+3 t -1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.024 (sec). Leaf size: 74

DSolve[{D[x[t],t]-2*x[t]-y[t]==2*Exp[t],D[y[t],t]-2*y[t]-4*z[t]==4*Exp[2*t],x[t]-D[z[t],t]-z[t]==0},{x[0]==9,y[0]==3,z[0]== 1},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to 3 t+2 e^t-3 e^{2 t}+8 e^{3 t}+2 \\ y(t)\to -6 t-4 e^t+8 e^{3 t}-1 \\ z(t)\to 3 t+e^t-e^{2 t}+2 e^{3 t}-1 \\ \end{align*}