6.21.3 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 : 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 {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*}

✓ Maple. Time used: 0.114 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 65

ode={D[ y1[t],t]==-4/5*y1[t]+3/5*y2[t],D[ y2[t],t]==-2/5*y1[t]-11/5*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.058 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(4*y__1(t)/5 - 3*y__2(t)/5 + Derivative(y__1(t), t),0),Eq(2*y__1(t)/5 + 11*y__2(t)/5 + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} - 3 C_{2} e^{- t}, \ y^{2}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{- t}\right ] \]