12.23.7 problem section 10.6, problem 7

Internal problem ID [2292]
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 7
Date solved : Monday, January 27, 2025 at 05:44:49 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 69

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

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

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 129

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

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