12.22.26 problem section 10.5, problem 26

Internal problem ID [2279]
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 26
Date solved : Monday, January 27, 2025 at 05:44:31 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 72

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

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= \frac {\left (2 c_3 \,t^{2}+4 c_2 t -c_3 t +4 c_1 \right ) {\mathrm e}^{-2 t}}{4} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (2 c_3 \,t^{2}+4 c_2 t +3 c_3 t +4 c_1 +4 c_2 +c_3 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 100

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

\begin{align*} \text {y1}(t)\to e^{-2 t} (c_1 (1-4 t)-4 (c_2+c_3) t) \\ \text {y2}(t)\to e^{-2 t} \left (-2 (c_1+c_2+c_3) t^2+(2 c_1+c_2+c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{-2 t} \left (2 (c_1+c_2+c_3) t^2+2 c_1 t+3 (c_2+c_3) t+c_3\right ) \\ \end{align*}