Internal problem ID [1623]
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 20.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (t )&=-7 y_{1}\relax (t )-4 y_{2}\relax (t )+4 y_{3}\relax (t )\\ y_{2}^{\prime }\relax (t )&=y_{1}\relax (t )+y_{3}\relax (t )\\ y_{3}^{\prime }\relax (t )&=-9 y_{1}\relax (t )-5 y_{2}\relax (t )+6 y_{3}\relax (t ) \end {align*}
With initial conditions \[ [y_{1}\relax (0) = -6, y_{2}\relax (0) = 9, y_{3}\relax (0) = -1] \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 80
dsolve([diff(y__1(t),t) = -7*y__1(t)-4*y__2(t)+4*y__3(t), diff(y__2(t),t) = y__1(t)+y__3(t), diff(y__3(t),t) = -9*y__1(t)-5*y__2(t)+6*y__3(t), y__1(0) = -6, y__2(0) = 9, y__3(0) = -1],[y__1(t), y__2(t), y__3(t)], singsol=all)
\[ y_{1}\relax (t ) = \frac {4 \,{\mathrm e}^{-t} \left (-\frac {17 \sin \left (2 t \right )}{2}-\frac {51 \cos \left (2 t \right )}{2}\right )}{17} \] \[ y_{2}\relax (t ) = \frac {9 \,{\mathrm e}^{t}}{2}-\frac {7 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}+\frac {9 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \] \[ y_{3}\relax (t ) = \frac {9 \,{\mathrm e}^{t}}{2}-\frac {7 \,{\mathrm e}^{-t} \sin \left (2 t \right )}{2}-\frac {11 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{2} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 50
DSolve[{y1'[t]==-7*y1[t]-4*y2[t]+4*y3[t],y2'[t]==-1*y1[t]-0*y2[t]+1*y3[t],y3'[t]==-9*y1[t]-5*y2[t]+6*y3[t]},{y1[0]==-6,y2[0]==9,y3[0]==-1},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to -2 e^{-3 t}-4 e^t \\ \text {y2}(t)\to e^t (9-4 t) \\ \text {y3}(t)\to e^t (1-4 t)-2 e^{-3 t} \\ \end{align*}