12.21.3 problem section 10.4, problem 3

Internal problem ID [2241]
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 3
Date solved : Monday, January 27, 2025 at 05:43:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-\frac {4 y_{1} \left (t \right )}{5}+\frac {3 y_{2} \left (t \right )}{5}\\ \frac {d}{d t}y_{2} \left (t \right )&=-\frac {2 y_{1} \left (t \right )}{5}-\frac {11 y_{2} \left (t \right )}{5} \end{align*}

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 35

dsolve([diff(y__1(t),t)=-4/5*y__1(t)+3/5*y__2(t),diff(y__2(t),t)=-2/5*y__1(t)-11/5*y__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 65

DSolve[{D[ y1[t],t]==-4/5*y1[t]+3/5*y2[t],D[ y2[t],t]==-2/5*y1[t]-11/5*y2[t]},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{5} e^{-2 t} \left (c_1 \left (6 e^t-1\right )+3 c_2 \left (e^t-1\right )\right ) \\ \text {y2}(t)\to \frac {1}{5} e^{-2 t} \left (-2 c_1 \left (e^t-1\right )-c_2 \left (e^t-6\right )\right ) \\ \end{align*}