6.23.7 problem section 10.6, problem 7

Internal problem ID [2292]
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 7
Date solved : Tuesday, September 30, 2025 at 05:26:01 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.164 (sec). Leaf size: 69

ode:=[diff(y__1(t),t) = 2*y__1(t)+y__2(t)-y__3(t), diff(y__2(t),t) = y__2(t)+y__3(t), diff(y__3(t),t) = y__1(t)+y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{t} \cos \left (t \right )-c_3 \,{\mathrm e}^{t} \sin \left (t \right ) \\ y_{2} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}-c_2 \,{\mathrm e}^{t} \cos \left (t \right )+c_3 \,{\mathrm e}^{t} \sin \left (t \right ) \\ y_{3} \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{t} \sin \left (t \right )+c_3 \,{\mathrm e}^{t} \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 129

ode={D[ y1[t],t]==2*y1[t]+1*y2[t]-1*y3[t],D[ y2[t],t]==0*y1[t]+1*y2[t]+1*y3[t],D[ y3[t],t]==1*y1[t]+0*y2[t]+1*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \frac {1}{2} e^t \left (-2 c_3 \sin (t)+c_2 \left (e^t+\sin (t)-\cos (t)\right )+c_1 \left (e^t+\sin (t)+\cos (t)\right )\right )\\ \text {y2}(t)&\to \frac {1}{2} e^t \left ((c_1+c_2) e^t+(c_2-c_1) \cos (t)-(c_1+c_2-2 c_3) \sin (t)\right )\\ \text {y3}(t)&\to \frac {1}{2} e^t \left ((c_1+c_2) e^t-(c_1+c_2-2 c_3) \cos (t)+(c_1-c_2) \sin (t)\right ) \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-2*y__1(t) - y__2(t) + y__3(t) + Derivative(y__1(t), t),0),Eq(-y__2(t) - y__3(t) + Derivative(y__2(t), t),0),Eq(-y__1(t) - y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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