12.21.2 problem section 10.4, problem 2

Internal problem ID [2240]
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 2
Date solved : Monday, January 27, 2025 at 05:43:58 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 76

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

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