12.22.5 problem section 10.5, problem 5

Internal problem ID [2258]
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 5
Date solved : Tuesday, March 04, 2025 at 01:52:34 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=4 y_{1} \left (t \right )+12 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-3 y_{1} \left (t \right )-8 y_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 34

ode:=[diff(y__1(t),t) = 4*y__1(t)+12*y__2(t), diff(y__2(t),t) = -3*y__1(t)-8*y__2(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (c_2 t +c_1 \right ) \\ y_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \left (6 c_2 t +6 c_1 -c_2 \right )}{12} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[ y1[t],t]==4*y1[t]+12*y2[t],D[ y2[t],t]==-3*y1[t]-8*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-2 t} (6 c_1 t+12 c_2 t+c_1) \\ \text {y2}(t)\to e^{-2 t} (c_2-3 (c_1+2 c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-4*y__1(t) - 12*y__2(t) + Derivative(y__1(t), t),0),Eq(3*y__1(t) + 8*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = 6 C_{1} t e^{- 2 t} + \left (C_{1} + 6 C_{2}\right ) e^{- 2 t}, \ y^{2}{\left (t \right )} = - 3 C_{1} t e^{- 2 t} - 3 C_{2} e^{- 2 t}\right ] \]