12.22.22 problem section 10.5, problem 22

Internal problem ID [2275]
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 22
Date solved : Tuesday, March 04, 2025 at 01:52:52 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=4 y_{1} \left (t \right )-8 y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-3 y_{1} \left (t \right )-y_{2} \left (t \right )-4 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=y_{1} \left (t \right )-y_{2} \left (t \right )+9 y_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = -4\\ y_{2} \left (0\right ) = 1\\ y_{3} \left (0\right ) = -3 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 61

ode:=[diff(y__1(t),t) = 4*y__1(t)-8*y__2(t)-4*y__3(t), diff(y__2(t),t) = -3*y__1(t)-y__2(t)-4*y__3(t), diff(y__3(t),t) = y__1(t)-y__2(t)+9*y__3(t)]; 
ic:=y__1(0) = -4y__2(0) = 1y__3(0) = -3; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (t \right ) &= \frac {22 \,{\mathrm e}^{9 t}}{13}-\frac {50 \,{\mathrm e}^{7 t}}{11}-\frac {164 \,{\mathrm e}^{-4 t}}{143} \\ y_{2} \left (t \right ) &= \frac {22 \,{\mathrm e}^{9 t}}{13}+\frac {5 \,{\mathrm e}^{7 t}}{11}-\frac {164 \,{\mathrm e}^{-4 t}}{143} \\ y_{3} \left (t \right ) &= -\frac {11 \,{\mathrm e}^{9 t}}{2}+\frac {5 \,{\mathrm e}^{7 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 57

ode={D[ y1[t],t]==4*y1[t]-8*y2[t]-4*y3[t],D[ y2[t],t]==-3*y1[t]-1*y2[t]-3*y3[t],D[ y3[t],t]==1*y1[t]-1*y2[t]+9*y3[t]}; 
ic={y1[0]==-4,y2[0]==1,y3[0]==-3}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{8 t} (8 t-3)-e^{-4 t} \\ \text {y2}(t)\to 2 e^{8 t}-e^{-4 t} \\ \text {y3}(t)\to -e^{8 t} (8 t+3) \\ \end{align*}

✓ Sympy. Time used: 0.146 (sec). Leaf size: 73

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-4*y__1(t) + 8*y__2(t) + 4*y__3(t) + Derivative(y__1(t), t),0),Eq(3*y__1(t) + y__2(t) + 4*y__3(t) + Derivative(y__2(t), t),0),Eq(-y__1(t) + y__2(t) - 9*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = C_{1} e^{- 4 t} - \frac {20 C_{2} e^{7 t}}{11} - \frac {4 C_{3} e^{9 t}}{13}, \ y^{2}{\left (t \right )} = C_{1} e^{- 4 t} + \frac {2 C_{2} e^{7 t}}{11} - \frac {4 C_{3} e^{9 t}}{13}, \ y^{3}{\left (t \right )} = C_{2} e^{7 t} + C_{3} e^{9 t}\right ] \]