12.23.8 problem section 10.6, problem 8

Internal problem ID [2293]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.6, constant coefficient homogeneous system III. Page 566
Problem number : section 10.6, problem 8
Date solved : Monday, January 27, 2025 at 05:44:50 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 132

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

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 163

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

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