12.22.17 problem section 10.5, problem 17

Internal problem ID [2270]
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 17
Date solved : Tuesday, March 04, 2025 at 01:52:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-7 y_{1} \left (t \right )+3 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-3 y_{1} \left (t \right )-y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 0\\ y_{2} \left (0\right ) = 2 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 25

ode:=[diff(y__1(t),t) = -7*y__1(t)+3*y__2(t), diff(y__2(t),t) = -3*y__1(t)-y__2(t)]; 
ic:=y__1(0) = 0y__2(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= 6 \,{\mathrm e}^{-4 t} t \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-4 t} \left (18 t +6\right )}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 27

ode={D[ y1[t],t]==-7*y1[t]+3*y2[t],D[ y2[t],t]==-3*y1[t]-1*y2[t]}; 
ic={y1[0]==0,y2[0]==2}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to 6 e^{-4 t} t \\ \text {y2}(t)\to e^{-4 t} (6 t+2) \\ \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(7*y__1(t) - 3*y__2(t) + Derivative(y__1(t), t),0),Eq(3*y__1(t) + y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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