12.22.9 problem section 10.5, problem 9

Internal problem ID [2262]
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 9
Date solved : Tuesday, March 04, 2025 at 01:52:38 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 74

ode:=[diff(y__1(t),t) = 1/3*y__1(t)+1/3*y__2(t)-y__3(t), diff(y__2(t),t) = -4/3*y__1(t)-4/3*y__2(t)+y__3(t), diff(y__3(t),t) = -2/3*y__1(t)+1/3*y__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 136

ode={D[ y1[t],t]==1/3*y1[t]+1/3*y2[t]-1*y3[t],D[ y2[t],t]==-4/3*y1[t]-4/3*y2[t]+1*y3[t],D[ y3[t],t]==-2/3*y1[t]+1/3*y2[t]+0*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (2 t+3 e^{2 t}+3\right )+2 c_2 t-3 c_3 \left (e^{2 t}-1\right )\right ) \\ \text {y2}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (-2 t-3 e^{2 t}+3\right )-2 c_2 (t-3)+3 c_3 \left (e^{2 t}-1\right )\right ) \\ \text {y3}(t)\to \frac {1}{6} e^{-t} \left (c_1 \left (2 t-3 e^{2 t}+3\right )+2 c_2 t+3 c_3 \left (e^{2 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.166 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__1(t)/3 - y__2(t)/3 + y__3(t) + Derivative(y__1(t), t),0),Eq(4*y__1(t)/3 + 4*y__2(t)/3 - y__3(t) + Derivative(y__2(t), t),0),Eq(2*y__1(t)/3 - y__2(t)/3 + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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