Internal problem ID [1641]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 10 Linear system of Differential equations. Section 10.6, constant coefficient
homogeneous system III. Page 566
Problem number: section 10.6, problem 6.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (t )&=-3 y_{1} \relax (t )+3 y_{2} \relax (t )+y_{3} \relax (t )\\ y_{2}^{\prime }\relax (t )&=y_{1} \relax (t )-5 y_{2} \relax (t )-3 y_{3} \relax (t )\\ y_{3}^{\prime }\relax (t )&=-3 y_{1} \relax (t )+7 y_{2} \relax (t )+3 y_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 142
dsolve([diff(y__1(t),t)=-3*y__1(t)+3*y__2(t)+1*y__3(t),diff(y__2(t),t)=1*y__1(t)-5*y__2(t)-3*y__3(t),diff(y__3(t),t)=-3*y__1(t)+7*y__2(t)+3*y__3(t)],[y__1(t), y__2(t), y__3(t)], singsol=all)
\[ y_{1} \relax (t ) = -{\mathrm e}^{-t} c_{1}+\frac {c_{2} {\mathrm e}^{-2 t} \sin \left (2 t \right )}{2}+\frac {c_{2} {\mathrm e}^{-2 t} \cos \left (2 t \right )}{2}+\frac {c_{3} {\mathrm e}^{-2 t} \cos \left (2 t \right )}{2}-\frac {c_{3} {\mathrm e}^{-2 t} \sin \left (2 t \right )}{2} \] \[ y_{2} \relax (t ) = -{\mathrm e}^{-t} c_{1}+\frac {c_{2} {\mathrm e}^{-2 t} \cos \left (2 t \right )}{2}-\frac {c_{3} {\mathrm e}^{-2 t} \sin \left (2 t \right )}{2}-\frac {c_{2} {\mathrm e}^{-2 t} \sin \left (2 t \right )}{2}-\frac {c_{3} {\mathrm e}^{-2 t} \cos \left (2 t \right )}{2} \] \[ y_{3} \relax (t ) = {\mathrm e}^{-t} c_{1}+c_{2} {\mathrm e}^{-2 t} \sin \left (2 t \right )+c_{3} {\mathrm e}^{-2 t} \cos \left (2 t \right ) \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 158
DSolve[{y1'[t]==-3*y1[t]+3*y2[t]+1*y3[t],y2'[t]==1*y1[t]-5*y2[t]-3*y3[t],y3'[t]==-3*y1[t]+7*y2[t]+3*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to e^{-2 t} \left ((c_1-c_2-c_3) e^t+(c_2+c_3) \cos (2 t)+(-c_1+2 c_2+c_3) \sin (2 t)\right ) \\ \text {y2}(t)\to e^{-2 t} \left ((c_1-c_2-c_3) e^t+(-c_1+2 c_2+c_3) \cos (2 t)-(c_2+c_3) \sin (2 t)\right ) \\ \text {y3}(t)\to e^{-2 t} \left ((-c_1+c_2+c_3) e^t+(c_1-c_2) \cos (2 t)+(-c_1+3 c_2+2 c_3) \sin (2 t)\right ) \\ \end{align*}