12.22.23 problem section 10.5, problem 23

Internal problem ID [2276]
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 23
Date solved : Tuesday, March 04, 2025 at 01:52:53 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=-5 y_{1} \left (t \right )-y_{2} \left (t \right )+11 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-7 y_{1} \left (t \right )+y_{2} \left (t \right )+13 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-4 y_{1} \left (t \right )+8 y_{3} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 43

ode:=[diff(y__1(t),t) = -5*y__1(t)-y__2(t)+11*y__3(t), diff(y__2(t),t) = -7*y__1(t)+y__2(t)+13*y__3(t), diff(y__3(t),t) = -4*y__1(t)+8*y__3(t)]; 
ic:=y__1(0) = 0y__2(0) = 2y__3(0) = 2; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= -3+3 \,{\mathrm e}^{4 t}+8 t \\ y_{2} \left (t \right ) &= 6 \,{\mathrm e}^{4 t}-4+4 t \\ y_{3} \left (t \right ) &= -1+4 t +3 \,{\mathrm e}^{4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 47

ode={D[ y1[t],t]==-5*y1[t]-1*y2[t]+11*y3[t],D[ y2[t],t]==-7*y1[t]+1*y2[t]+13*y3[t],D[ y3[t],t]==-4*y1[t]-0*y2[t]+8*y3[t]}; 
ic={y1[0]==0,y2[0]==2,y3[0]==2}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to 8 t+3 e^{4 t}-3 \\ \text {y2}(t)\to 4 t+6 e^{4 t}-4 \\ \text {y3}(t)\to 4 t+3 e^{4 t}-1 \\ \end{align*}

✓ Sympy. Time used: 0.126 (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(5*y__1(t) + y__2(t) - 11*y__3(t) + Derivative(y__1(t), t),0),Eq(7*y__1(t) - y__2(t) - 13*y__3(t) + Derivative(y__2(t), t),0),Eq(4*y__1(t) - 8*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = - \frac {8 C_{1}}{3} - \frac {8 C_{2} t}{3} + \frac {C_{2}}{3} + C_{3} e^{4 t}, \ y^{2}{\left (t \right )} = - \frac {4 C_{1}}{3} - \frac {4 C_{2} t}{3} + C_{2} + 2 C_{3} e^{4 t}, \ y^{3}{\left (t \right )} = - \frac {4 C_{1}}{3} - \frac {4 C_{2} t}{3} + C_{3} e^{4 t}\right ] \]