76.28.1 problem 2

Internal problem ID [17868]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.6 (Nonhomogeneous Linear Systems). Problems at page 436
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 11:04:16 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.093 (sec). Leaf size: 53

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

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

✓ Solution by Mathematica

Time used: 0.213 (sec). Leaf size: 97

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

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t} \left (4 e^t t+e^{2 t} (6 t-1+6 c_1-2 c_2)-2 c_1+2 c_2\right ) \\ \text {x2}(t)\to \frac {1}{4} e^{-t} \left (e^t (8 t-4)+e^{2 t} (6 t-3+6 c_1-2 c_2)-6 c_1+6 c_2\right ) \\ \end{align*}