12.21.5 problem section 10.4, problem 5

Internal problem ID [2243]
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 5
Date solved : Tuesday, March 04, 2025 at 01:52:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=2 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: 0.020 (sec). Leaf size: 34

ode:=[diff(y__1(t),t) = 2*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 ) &= c_1 \,{\mathrm e}^{3 t}+{\mathrm e}^{-2 t} c_2 \\ y_{2} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{3 t}}{4}+{\mathrm e}^{-2 t} c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 72

ode={D[ y1[t],t]==2*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}{5} e^{-2 t} \left (c_1 \left (4 e^{5 t}+1\right )-4 c_2 \left (e^{5 t}-1\right )\right ) \\ \text {y2}(t)\to \frac {1}{5} e^{-2 t} \left (c_2 \left (e^{5 t}+4\right )-c_1 \left (e^{5 t}-1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-2*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)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} e^{- 2 t} - 4 C_{2} e^{3 t}, \ y^{2}{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{3 t}\right ] \]