76.23.2 problem 3

Internal problem ID [17793]
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 6.1 (Definitions and examples). Problems at page 388
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 11:03:10 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.122 (sec). Leaf size: 66

dsolve([diff(x__1(t),t)=x__1(t)-x__2(t)+4*x__3(t),diff(x__2(t),t)=3*x__1(t)+2*x__2(t)-x__3(t),diff(x__3(t),t)=2*x__1(t)+x__2(t)-x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 182

DSolve[{D[x1[t],t]==x1[t]-x2[t]+4*x3[t],D[x2[t],t]==3*x1[t]+2*x2[t]-x3[t],D[x3[t],t]==2*x1[t]+x2[t]-x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (e^{3 t}+3 e^{5 t}+2\right )-2 c_2 \left (e^{3 t}-1\right )+3 c_3 \left (e^{3 t}+e^{5 t}-2\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-2 t} \left (c_1 \left (-2 e^{3 t}+3 e^{5 t}-1\right )+c_2 \left (4 e^{3 t}-1\right )+3 c_3 \left (-2 e^{3 t}+e^{5 t}+1\right )\right ) \\ \text {x3}(t)\to \frac {1}{6} e^{-2 t} \left (c_1 \left (-e^{3 t}+3 e^{5 t}-2\right )+2 c_2 \left (e^{3 t}-1\right )+3 c_3 \left (-e^{3 t}+e^{5 t}+2\right )\right ) \\ \end{align*}