14.26.3 problem 3

Internal problem ID [2776]
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 : 3
Date solved : Monday, January 27, 2025 at 06:13:11 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 48

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

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{3}-\frac {4 \,{\mathrm e}^{-t}}{3}+3 \,{\mathrm e}^{t}-t \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{6}-\frac {8 \,{\mathrm e}^{-t}}{3}+3 \,{\mathrm e}^{t}+\frac {1}{2}-t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.412 (sec). Leaf size: 59

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

\begin{align*} \text {x1}(t)\to \frac {1}{3} \left (-3 t-4 e^{-t}+9 e^t+e^{2 t}\right ) \\ \text {x2}(t)\to \frac {1}{6} \left (-6 t-16 e^{-t}+18 e^t+e^{2 t}+3\right ) \\ \end{align*}