12.21.12 problem section 10.4, problem 12

Internal problem ID [2250]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 12
Date solved : Monday, January 27, 2025 at 05:44:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 70

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

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{3 t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-2 t} \\ y_{2} \left (t \right ) &= \frac {7 c_1 \,{\mathrm e}^{3 t}}{11}+c_2 \,{\mathrm e}^{-t}+2 c_3 \,{\mathrm e}^{-2 t} \\ y_{3} \left (t \right ) &= \frac {c_1 \,{\mathrm e}^{3 t}}{11}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{-2 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 102

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

\begin{align*} \text {y1}(t)\to \frac {1}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right ) \\ \text {y2}(t)\to \frac {1}{216} e^{-2 t} \left (2 c_1 \left (27 e^t-8\right )+c_2 \left (16-27 e^t\right )\right ) \\ \text {y3}(t)\to \frac {1}{216} e^{-2 t} \left (c_1 \left (54 e^t-8\right )+c_2 \left (8-27 e^t\right )\right ) \\ \end{align*}