12.22.19 problem section 10.5, problem 19

Internal problem ID [2272]
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 19
Date solved : Saturday, March 29, 2025 at 11:52:23 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = -6\\ y_{2} \left (0\right ) = -2\\ y_{3} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 40

ode:=[diff(y__1(t),t) = -2*y__1(t)+2*y__2(t)+y__3(t), diff(y__2(t),t) = -2*y__1(t)+2*y__2(t)+y__3(t), diff(y__3(t),t) = -3*y__1(t)+3*y__2(t)+2*y__3(t)]; 
ic:=y__1(0) = -6y__2(0) = -2y__3(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= -9+2 t +3 \,{\mathrm e}^{2 t} \\ y_{2} \left (t \right ) &= 3 \,{\mathrm e}^{2 t}-5+2 t \\ y_{3} \left (t \right ) &= -6+6 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 44

ode={D[ y1[t],t]==-2*y1[t]+2*y2[t]+1*y3[t],D[ y2[t],t]==-2*y1[t]+2*y2[t]+1*y3[t],D[ y3[t],t]==-3*y1[t]+3*y2[t]+2*y3[t]}; 
ic={y1[0]==-6,y2[0]==-2,y3[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to 2 t+3 e^{2 t}-9 \\ \text {y2}(t)\to 2 t+3 e^{2 t}-5 \\ \text {y3}(t)\to 6 \left (e^{2 t}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 53

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) - 2*y__2(t) - y__3(t) + Derivative(y__1(t), t),0),Eq(2*y__1(t) - 2*y__2(t) - y__3(t) + Derivative(y__2(t), t),0),Eq(3*y__1(t) - 3*y__2(t) - 2*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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