12.22.10 problem section 10.5, problem 10

Internal problem ID [2263]
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 10
Date solved : Monday, January 27, 2025 at 05:44:17 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-2 y_{1} \left (t \right )+2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-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: 62

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 125

DSolve[{D[ y1[t],t]==-1*y1[t]+1*y2[t]-1*y3[t],D[ y2[t],t]==-2*y1[t]+0*y2[t]+2*y3[t],D[ y3[t],t]==-1*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 (t+1)+(c_2-c_3) t) \\ \text {y2}(t)\to \frac {1}{2} e^{-2 t} \left (-\left (c_1 \left (e^{4 t}-1\right )\right )+c_2 \left (e^{4 t}+1\right )+c_3 \left (e^{4 t}-1\right )\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left (c_1 \left (2 t-e^{4 t}+1\right )+c_2 \left (2 t+e^{4 t}-1\right )+c_3 \left (-2 t+e^{4 t}+1\right )\right ) \\ \end{align*}