12.22.21 problem section 10.5, problem 21

Internal problem ID [2274]
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 21
Date solved : Monday, January 27, 2025 at 05:44:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-y_{1} \left (t \right )-4 y_{2} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=3 y_{1} \left (t \right )+6 y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-3 y_{1} \left (t \right )-2 y_{2} \left (t \right )+3 y_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = -2\\ y_{2} \left (0\right ) = 1\\ y_{3} \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 62

dsolve([diff(y__1(t),t) = -y__1(t)-4*y__2(t)-y__3(t), diff(y__2(t),t) = 3*y__1(t)+6*y__2(t)+y__3(t), diff(y__3(t),t) = -3*y__1(t)-2*y__2(t)+3*y__3(t), y__1(0) = -2, y__2(0) = 1, y__3(0) = 3], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= -2 \,{\mathrm e}^{4 t}+3 \,{\mathrm e}^{2 t} t \\ y_{2} \left (t \right ) &= 2 \,{\mathrm e}^{4 t}-3 \,{\mathrm e}^{2 t} t -{\mathrm e}^{2 t} \\ y_{3} \left (t \right ) &= 2 \,{\mathrm e}^{4 t}+3 \,{\mathrm e}^{2 t} t +{\mathrm e}^{2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 63

DSolve[{D[ y1[t],t]==-1*y1[t]-4*y2[t]-1*y3[t],D[ y2[t],t]==3*y1[t]+6*y2[t]+1*y3[t],D[ y3[t],t]==-3*y1[t]-2*y2[t]+3*y3[t]},{y1[0]==-2,y2[0]==1,y3[0]==3},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to 3 e^{2 t} t-2 e^{4 t} \\ \text {y2}(t)\to e^{2 t} \left (-3 t+2 e^{2 t}-1\right ) \\ \text {y3}(t)\to e^{2 t} \left (3 t+2 e^{2 t}+1\right ) \\ \end{align*}