7.25.19 problem 19

Internal problem ID [639]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 19
Date solved : Monday, January 27, 2025 at 02:56:35 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 63

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

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t}+c_1 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= -2 c_2 \,{\mathrm e}^{3 t}+c_3 \,{\mathrm e}^{6 t}-c_1 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 124

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

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