12.22.30 problem section 10.5, problem 30

Internal problem ID [2283]
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 30
Date solved : Monday, January 27, 2025 at 05:44:34 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-4 y_{1} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-y_{1} \left (t \right )-3 y_{2} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=y_{1} \left (t \right )-2 y_{3} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 43

dsolve([diff(y__1(t),t)=-4*y__1(t)-0*y__2(t)-1*y__3(t),diff(y__2(t),t)=-1*y__1(t)-3*y__2(t)-1*y__3(t),diff(y__3(t),t)=1*y__1(t)-0*y__2(t)-2*y__3(t)],singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_3 t +c_1 +c_2 \right ) \\ y_{3} \left (t \right ) &= -{\mathrm e}^{-3 t} \left (c_3 t +c_2 +c_3 \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 63

DSolve[{D[ y1[t],t]==-4*y1[t]-0*y2[t]-1*y3[t],D[ y2[t],t]==-1*y1[t]-3*y2[t]-1*y3[t],D[ y3[t],t]==1*y1[t]-0*y2[t]-2*y3[t]},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to e^{-3 t} (c_1 (-t)-c_3 t+c_1) \\ \text {y2}(t)\to e^{-3 t} (c_2-(c_1+c_3) t) \\ \text {y3}(t)\to e^{-3 t} ((c_1+c_3) t+c_3) \\ \end{align*}