12.22.31 problem section 10.5, problem 31

Internal problem ID [2284]
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 31
Date solved : Tuesday, March 04, 2025 at 01:53:01 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 52

ode:=[diff(y__1(t),t) = -3*y__1(t)-3*y__2(t)+4*y__3(t), diff(y__2(t),t) = 4*y__1(t)+5*y__2(t)-8*y__3(t), diff(y__3(t),t) = 2*y__1(t)+3*y__2(t)-5*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} \left (c_3 t +c_2 \right ) \\ y_{2} \left (t \right ) &= {\mathrm e}^{-t} \left (-2 c_3 t +c_1 -2 c_2 \right ) \\ y_{3} \left (t \right ) &= \frac {{\mathrm e}^{-t} \left (-4 c_3 t +3 c_1 -4 c_2 +c_3 \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 83

ode={D[ y1[t],t]==-3*y1[t]-3*y2[t]+4*y3[t],D[ y2[t],t]==4*y1[t]+5*y2[t]-8*y3[t],D[ y3[t],t]==2*y1[t]+3*y2[t]-5*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to e^{-t} (-2 c_1 t-3 c_2 t+4 c_3 t+c_1) \\ \text {y2}(t)\to e^{-t} ((4 c_1+6 c_2-8 c_3) t+c_2) \\ \text {y3}(t)\to e^{-t} ((2 c_1+3 c_2-4 c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(3*y__1(t) + 3*y__2(t) - 4*y__3(t) + Derivative(y__1(t), t),0),Eq(-4*y__1(t) - 5*y__2(t) + 8*y__3(t) + Derivative(y__2(t), t),0),Eq(-2*y__1(t) - 3*y__2(t) + 5*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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