14.26.5 problem 5

Internal problem ID [2778]
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 : 5
Date solved : Monday, January 27, 2025 at 06:13:12 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 31

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

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