12.21.6 problem section 10.4, problem 6

Internal problem ID [2244]
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 6
Date solved : Tuesday, March 04, 2025 at 01:52:20 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 30

ode:=[diff(y__1(t),t) = 4*y__1(t)-3*y__2(t), diff(y__2(t),t) = 2*y__1(t)-y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 56

ode={D[ y1[t],t]==4*y1[t]-3*y2[t],D[ y2[t],t]==2*y1[t]-1*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.080 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-4*y__1(t) + 3*y__2(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) + y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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