14.26.11 problem 11

Internal problem ID [2784]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 11
Date solved : Monday, January 27, 2025 at 06:13:49 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+2 x_{3} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )-x_{2} \left (t \right )+4 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 32

dsolve([diff(x__1(t),t) = x__1(t)+2*x__2(t)-3*x__3(t), diff(x__2(t),t) = x__1(t)+x__2(t)+2*x__3(t), diff(x__3(t),t) = x__1(t)-x__2(t)+4*x__3(t), x__1(0) = 1, x__2(0) = 0, x__3(0) = 0], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (-t +1\right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{2 t} t \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 35

DSolve[{D[x1[t],t]==1*x1[t]+2*x2[t]-3*x3[t],D[x2[t],t]==1*x1[t]+1*x2[t]+2*x3[t],D[x3[t],t]==1*x1[t]-1*x2[t]+4*x3[t]},{x1[0]==1,x2[0]==0,x3[0]==0},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to -e^{2 t} (t-1) \\ \text {x2}(t)\to e^{2 t} t \\ \text {x3}(t)\to e^{2 t} t \\ \end{align*}