6.21.2 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 : Tuesday, September 30, 2025 at 05:25:21 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*}

✓ Maple. Time used: 0.117 (sec). Leaf size: 34

ode:=[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)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 76

ode={D[ y1[t],t]==-5/4*y1[t]+3/4*y2[t],D[ y2[t],t]==3/4*y1[t]-5/4*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(5*y__1(t)/4 - 3*y__2(t)/4 + Derivative(y__1(t), t),0),Eq(-3*y__1(t)/4 + 5*y__2(t)/4 + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - C_{1} e^{- 2 t} + C_{2} e^{- \frac {t}{2}}, \ y^{2}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- \frac {t}{2}}\right ] \]