12.23.4 problem section 10.6, problem 4

Internal problem ID [2289]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.6, constant coefficient homogeneous system III. Page 566
Problem number : section 10.6, problem 4
Date solved : Tuesday, March 04, 2025 at 01:53:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=5 y_{1} \left (t \right )-6 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*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 57

ode:=[diff(y__1(t),t) = 5*y__1(t)-6*y__2(t), diff(y__2(t),t) = 3*y__1(t)-y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 60

ode={D[ y1[t],t]==5*y1[t]-6*y2[t],D[ y2[t],t]==3*y1[t]-1*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.117 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-5*y__1(t) + 6*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)
 

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