6.23.3 problem section 10.6, problem 3

Internal problem ID [2288]
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 3
Date solved : Tuesday, September 30, 2025 at 05:25:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=y_{1} \left (t \right )+2 y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-4 y_{1} \left (t \right )+5 y_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.129 (sec). Leaf size: 56

ode:=[diff(y__1(t),t) = y__1(t)+2*y__2(t), diff(y__2(t),t) = -4*y__1(t)+5*y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

ode={D[ y1[t],t]==1*y1[t]+2*y2[t],D[ y2[t],t]==-4*y1[t]+5*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.072 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-y__1(t) - 2*y__2(t) + Derivative(y__1(t), t),0),Eq(4*y__1(t) - 5*y__2(t) + Derivative(y__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t)],ics=ics)
 

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