9.4.23 problem problem 23

Internal problem ID [987]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 23
Date solved : Monday, January 27, 2025 at 03:22:44 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-5 x_{1} \left (t \right )-3 x_{2} \left (t \right )-x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )+5 x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 58

dsolve([diff(x__1(t),t)=3*x__1(t)+1*x__2(t)+1*x__3(t),diff(x__2(t),t)=-5*x__1(t)-3*x__2(t)-1*x__3(t),diff(x__3(t),t)=5*x__1(t)+5*x__2(t)+3*x__3(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 121

DSolve[{D[ x1[t],t]==3*x1[t]+1*x2[t]+1*x3[t],D[ x2[t],t]==-5*x1[t]-3*x2[t]-1*x3[t],D[ x3[t],t]==5*x1[t]+5*x2[t]+3*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

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