12.22.18 problem section 10.5, problem 18

Internal problem ID [2271]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.5, constant coefficient homogeneous system II. Page 555
Problem number : section 10.5, problem 18
Date solved : Monday, January 27, 2025 at 05:44:23 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 6\\ y_{2} \left (0\right ) = 5\\ y_{3} \left (0\right ) = -7 \end{align*}

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 56

dsolve([diff(y__1(t),t) = -y__1(t)+y__2(t), diff(y__2(t),t) = y__1(t)-y__2(t)-2*y__3(t), diff(y__3(t),t) = -y__1(t)-y__2(t)-y__3(t), y__1(0) = 6, y__2(0) = 5, y__3(0) = -7], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= 4 \,{\mathrm e}^{t}+2 \,{\mathrm e}^{-2 t}-{\mathrm e}^{-2 t} t \\ y_{2} \left (t \right ) &= 8 \,{\mathrm e}^{t}-3 \,{\mathrm e}^{-2 t}+{\mathrm e}^{-2 t} t \\ y_{3} \left (t \right ) &= -6 \,{\mathrm e}^{t}-{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 58

DSolve[{D[ y1[t],t]==-1*y1[t]+1*y2[t]+0*y3[t],D[ y2[t],t]==1*y1[t]-1*y2[t]-2*y3[t],D[ y3[t],t]==-1*y1[t]-1*y2[t]-1*y3[t]},{y1[0]==6,y2[0]==5,y3[0]==-7},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to e^{-2 t} \left (-t+4 e^{3 t}+2\right ) \\ \text {y2}(t)\to e^{-2 t} \left (t+8 e^{3 t}-3\right ) \\ \text {y3}(t)\to -e^{-2 t}-6 e^t \\ \end{align*}