12.21.4 problem section 10.4, problem 4

Internal problem ID [2242]
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 4
Date solved : Tuesday, March 04, 2025 at 01:52:18 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 31

ode:=[diff(y__1(t),t) = -y__1(t)-4*y__2(t), diff(y__2(t),t) = -y__1(t)-y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

ode={D[ y1[t],t]==-1*y1[t]-4*y2[t],D[ y2[t],t]==-1*y1[t]-1*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{2} e^{-3 t} \left (c_1 \left (e^{4 t}+1\right )-2 c_2 \left (e^{4 t}-1\right )\right ) \\ \text {y2}(t)\to \frac {1}{4} e^{-3 t} \left (2 c_2 \left (e^{4 t}+1\right )-c_1 \left (e^{4 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 31

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

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