14.26.12 problem 12

Internal problem ID [2785]
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 : 12
Date solved : Monday, January 27, 2025 at 06:13:50 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 40

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

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

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 42

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

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