12.23.1 problem section 10.6, problem 1

Internal problem ID [2286]
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 1
Date solved : Monday, January 27, 2025 at 05:44:36 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 47

dsolve([diff(y__1(t),t)=-1*y__1(t)+2*y__2(t),diff(y__2(t),t)=-5*y__1(t)+5*y__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 55

DSolve[{D[ y1[t],t]==-1*y1[t]+2*y2[t],D[ y2[t],t]==-5*y1[t]+5*y2[t]},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to e^{2 t} (c_1 \cos (t)+(2 c_2-3 c_1) \sin (t)) \\ \text {y2}(t)\to e^{2 t} (c_2 (3 \sin (t)+\cos (t))-5 c_1 \sin (t)) \\ \end{align*}