14.25.7 problem 5

Internal problem ID [2764]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 5
Date solved : Monday, January 27, 2025 at 06:12:58 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 )&=x_{2} \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: 0.066 (sec). Leaf size: 44

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) = x__2(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 ) &= \left (-t -2\right ) {\mathrm e}^{2 t}+3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= {\mathrm e}^{2 t} \\ x_{3} \left (t \right ) &= -2 \,{\mathrm e}^{2 t}+3 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 45

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]+1*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 (-t+3 e^t-2\right ) \\ \text {x2}(t)\to e^{2 t} \\ \text {x3}(t)\to e^{2 t} \left (3 e^t-2\right ) \\ \end{align*}