Internal problem ID [1642]
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 7.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (t )&=2 y_{1} \relax (t )+y_{2} \relax (t )-y_{3} \relax (t )\\ y_{2}^{\prime }\relax (t )&=y_{2} \relax (t )+y_{3} \relax (t )\\ y_{3}^{\prime }\relax (t )&=y_{1} \relax (t )+y_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 70
dsolve([diff(y__1(t),t)=2*y__1(t)+1*y__2(t)-1*y__3(t),diff(y__2(t),t)=0*y__1(t)+1*y__2(t)+1*y__3(t),diff(y__3(t),t)=1*y__1(t)+0*y__2(t)+1*y__3(t)],[y__1(t), y__2(t), y__3(t)], singsol=all)
\[ y_{1} \relax (t ) = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{t} \cos \relax (t )-c_{3} {\mathrm e}^{t} \sin \relax (t ) \] \[ y_{2} \relax (t ) = c_{1} {\mathrm e}^{2 t}-c_{2} {\mathrm e}^{t} \cos \relax (t )+c_{3} {\mathrm e}^{t} \sin \relax (t ) \] \[ y_{3} \relax (t ) = c_{1} {\mathrm e}^{2 t}+c_{2} {\mathrm e}^{t} \sin \relax (t )+c_{3} {\mathrm e}^{t} \cos \relax (t ) \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 129
DSolve[{y1'[t]==2*y1[t]+1*y2[t]-1*y3[t],y2'[t]==0*y1[t]+1*y2[t]+1*y3[t],y3'[t]==1*y1[t]+0*y2[t]+1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to \frac {1}{2} e^t \left (-2 c_3 \sin (t)+c_2 \left (e^t+\sin (t)-\cos (t)\right )+c_1 \left (e^t+\sin (t)+\cos (t)\right )\right ) \\ \text {y2}(t)\to \frac {1}{2} e^t \left ((c_1+c_2) e^t+(c_2-c_1) \cos (t)-(c_1+c_2-2 c_3) \sin (t)\right ) \\ \text {y3}(t)\to \frac {1}{2} e^t \left ((c_1+c_2) e^t-(c_1+c_2-2 c_3) \cos (t)+(c_1-c_2) \sin (t)\right ) \\ \end{align*}