Internal problem ID [1612]
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 9.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} y_{1}^{\prime }\relax (t )&=\frac {y_{1}\relax (t )}{3}+\frac {y_{2}\relax (t )}{3}-y_{3}\relax (t )\\ y_{2}^{\prime }\relax (t )&=-\frac {4 y_{1}\relax (t )}{3}-\frac {4 y_{2}\relax (t )}{3}+y_{3}\relax (t )\\ y_{3}^{\prime }\relax (t )&=-\frac {2 y_{1}\relax (t )}{3}+\frac {y_{2}\relax (t )}{3} \end {align*}
✓ Solution by Maple
Time used: 0.057 (sec). Leaf size: 75
dsolve([diff(y__1(t),t)=1/3*y__1(t)+1/3*y__2(t)-1*y__3(t),diff(y__2(t),t)=-4/3*y__1(t)-4/3*y__2(t)+1*y__3(t),diff(y__3(t),t)=-2/3*y__1(t)+1/3*y__2(t)+0*y__3(t)],[y__1(t), y__2(t), y__3(t)], singsol=all)
\[ y_{1}\relax (t ) = -c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-t} t \] \[ y_{2}\relax (t ) = c_{1} {\mathrm e}^{t}-c_{2} {\mathrm e}^{-t}-c_{3} {\mathrm e}^{-t} t +3 c_{3} {\mathrm e}^{-t} \] \[ y_{3}\relax (t ) = c_{1} {\mathrm e}^{t}+c_{2} {\mathrm e}^{-t}+c_{3} {\mathrm e}^{-t} t \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 136
DSolve[{y1'[t]==1/3*y1[t]+1/3*y2[t]-1*y3[t],y2'[t]==-4/3*y1[t]-4/3*y2[t]+1*y3[t],y3'[t]==-2/3*y1[t]+1/3*y2[t]+0*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {y1}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (2 t+3 e^{2 t}+3\right )+2 c_2 t-3 c_3 \left (e^{2 t}-1\right )\right ) \\ \text {y2}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (-2 t-3 e^{2 t}+3\right )-2 c_2 (t-3)+3 c_3 \left (e^{2 t}-1\right )\right ) \\ \text {y3}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (2 t-3 e^{2 t}+3\right )+2 c_2 t+3 c_3 \left (e^{2 t}+1\right )\right ) \\ \end{align*}