12.21.9 problem section 10.4, problem 9

Internal problem ID [2247]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 9
Date solved : Monday, January 27, 2025 at 05:44:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-6 y_{1} \left (t \right )-4 y_{2} \left (t \right )-8 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-4 y_{1} \left (t \right )-4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-8 y_{1} \left (t \right )-4 y_{2} \left (t \right )-6 y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 66

dsolve([diff(y__1(t),t)=-6*y__1(t)-4*y__2(t)-8*y__3(t),diff(y__2(t),t)=-4*y__1(t)-0*y__2(t)-4*y__3(t),diff(y__3(t),t)=-8*y__1(t)-4*y__2(t)-6*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= 2 c_2 \,{\mathrm e}^{2 t}+2 c_3 \,{\mathrm e}^{-16 t}+{\mathrm e}^{2 t} c_1 \\ y_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{2 t}+c_3 \,{\mathrm e}^{-16 t} \\ y_{3} \left (t \right ) &= -\frac {5 c_2 \,{\mathrm e}^{2 t}}{2}+2 c_3 \,{\mathrm e}^{-16 t}-{\mathrm e}^{2 t} c_1 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 116

DSolve[{D[ y1[t],t]==-6*y1[t]-4*y2[t]-8*y3[t],D[ y2[t],t]==-4*y1[t]-0*y2[t]-4*y3[t],D[ y1[t],t]==-8*y1[t]-4*y2[t]-6*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {e^{-16 t} \left (c_1 \left (8-4913 e^{18 t}\right )+2 c_2 \left (4913 e^{18 t}+1\right )\right )}{44217} \\ \text {y2}(t)\to \frac {e^{-16 t} \left (4 c_1 \left (4913 e^{18 t}+1\right )+c_2 \left (1-39304 e^{18 t}\right )\right )}{44217} \\ \text {y3}(t)\to \frac {e^{-16 t} \left (c_1 \left (8-4913 e^{18 t}\right )+2 c_2 \left (4913 e^{18 t}+1\right )\right )}{44217} \\ \end{align*}