Internal problem ID [1621]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ y_{3}^{\prime }\left (t \right )&=-y_{1} \left (t \right )-y_{2} \left (t \right )-y_{3} \left (t \right ) \end {align*}
With initial conditions \[ [y_{1} \left (0\right ) = 6, y_{2} \left (0\right ) = 5, y_{3} \left (0\right ) = -7] \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 57
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[{y1'[t]==-1*y1[t]+1*y2[t]+0*y3[t],y2'[t]==1*y1[t]-1*y2[t]-2*y3[t],y3'[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*}