Internal problem ID [1600]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient
homogeneous system. Page 540
Problem number: section 10.4, problem 12.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (t )&=4 y_{1} \relax (t )-y_{2} \relax (t )-4 y_{3} \relax (t )\\ y_{2}^{\prime }\relax (t )&=4 y_{1} \relax (t )-3 y_{2} \relax (t )-2 y_{3} \relax (t )\\ y_{3}^{\prime }\relax (t )&=y_{1} \relax (t )-y_{2} \relax (t )-y_{3} \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 71
dsolve([diff(y__1(t),t)=4*y__1(t)-1*y__2(t)-4*y__3(t),diff(y__2(t),t)=4*y__1(t)-3*y__2(t)-2*y__3(t),diff(y__3(t),t)=1*y__1(t)-1*y__2(t)-1*y__3(t)],[y__1(t), y__2(t), y__3(t)], singsol=all)
\[ y_{1} \relax (t ) = c_{1} {\mathrm e}^{-2 t}+11 c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{-t} \] \[ y_{2} \relax (t ) = 2 c_{1} {\mathrm e}^{-2 t}+7 c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{-t} \] \[ y_{3} \relax (t ) = c_{1} {\mathrm e}^{-2 t}+c_{2} {\mathrm e}^{3 t}+c_{3} {\mathrm e}^{-t} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 102
DSolve[{y1'[t]==4*y1[t]-1*y2[t]-4*y3[t],y2'[t]==4*y1[t]-3*y2[t]-2*y3[t],y1'[t]==1*y1[t]-1*y2[t]-1*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to \frac {1}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right ) \\ \text {y2}(t)\to \frac {1}{216} e^{-2 t} \left (2 c_1 \left (27 e^t-8\right )+c_2 \left (16-27 e^t\right )\right ) \\ \text {y3}(t)\to \frac {1}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right ) \\ \end{align*}