12.21.4 problem section 10.4, problem 4

Internal problem ID [2242]
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 4
Date solved : Monday, January 27, 2025 at 05:44:00 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 71

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

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