12.22.2 problem section 10.5, problem 2

Internal problem ID [2255]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.5, constant coefficient homogeneous system II. Page 555
Problem number : section 10.5, problem 2
Date solved : Tuesday, March 04, 2025 at 01:52:31 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 30

ode:=[diff(y__1(t),t) = -y__2(t), diff(y__2(t),t) = y__1(t)-2*y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 44

ode={D[ y1[t],t]==0*y1[t]-1*y2[t],D[ y2[t],t]==1*y1[t]-2*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-t} (c_1 (t+1)-c_2 t) \\ \text {y2}(t)\to e^{-t} ((c_1-c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.073 (sec). Leaf size: 29

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

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