12.22.10 problem section 10.5, problem 10

Internal problem ID [2263]
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 10
Date solved : Tuesday, March 04, 2025 at 01:52:39 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 62

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 125

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

\begin{align*} \text {y1}(t)\to e^{-2 t} (c_1 (t+1)+(c_2-c_3) t) \\ \text {y2}(t)\to \frac {1}{2} e^{-2 t} \left (-\left (c_1 \left (e^{4 t}-1\right )\right )+c_2 \left (e^{4 t}+1\right )+c_3 \left (e^{4 t}-1\right )\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left (c_1 \left (2 t-e^{4 t}+1\right )+c_2 \left (2 t+e^{4 t}-1\right )+c_3 \left (-2 t+e^{4 t}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.152 (sec). Leaf size: 65

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) - y__2(t) + y__3(t) + Derivative(y__1(t), t),0),Eq(2*y__1(t) - 2*y__3(t) + Derivative(y__2(t), t),0),Eq(y__1(t) - 3*y__2(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)
 

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