12.22.8 problem section 10.5, problem 8

Internal problem ID [2261]
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 8
Date solved : Monday, January 27, 2025 at 05:44:15 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 70

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 131

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

\begin{align*} \text {y1}(t)\to \frac {1}{4} \left (c_1 \left (e^{4 t} (2-8 t)+2\right )+e^{4 t} (8 c_2 t+c_3)-c_3\right ) \\ \text {y2}(t)\to \frac {1}{4} \left (-2 c_1 \left (e^{4 t} (4 t+1)-1\right )+e^{4 t} (c_2 (8 t+4)+c_3)-c_3\right ) \\ \text {y3}(t)\to c_1 \left (e^{4 t} (1-4 t)-1\right )+\frac {1}{2} \left (e^{4 t} (8 c_2 t+c_3)+c_3\right ) \\ \end{align*}