12.21.14 problem section 10.4, problem 14

Internal problem ID [2252]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 10 Linear system of Differential equations. Section 10.4, constant coefficient homogeneous system. Page 540
Problem number : section 10.4, problem 14
Date solved : Tuesday, March 04, 2025 at 01:52:28 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=3 y_{1} \left (t \right )+2 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-2 y_{1} \left (t \right )+7 y_{2} \left (t \right )-2 y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-10 y_{1} \left (t \right )+10 y_{2} \left (t \right )-5 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 56

ode:=[diff(y__1(t),t) = 3*y__1(t)+2*y__2(t)-2*y__3(t), diff(y__2(t),t) = -2*y__1(t)+7*y__2(t)-2*y__3(t), diff(y__3(t),t) = -10*y__1(t)+10*y__2(t)-5*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{-5 t}+c_3 \,{\mathrm e}^{5 t} \\ y_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-5 t}+c_3 \,{\mathrm e}^{5 t}+c_1 \,{\mathrm e}^{5 t} \\ y_{3} \left (t \right ) &= 5 c_2 \,{\mathrm e}^{-5 t}+c_1 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 126

ode={D[ y1[t],t]==3*y1[t]+2*y2[t]-2*y3[t],D[ y2[t],t]==-2*y1[t]+7*y2[t]-2*y3[t],D[ y1[t],t]==-10*y1[t]+10*y2[t]-5*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to -\frac {e^{5 t} \left (c_1 \left (432 e^{10 t/3}-1331\right )+c_2 \left (1331-216 e^{10 t/3}\right )\right )}{85184} \\ \text {y2}(t)\to -\frac {e^{5 t} \left (c_1 \left (216 e^{10 t/3}-1331\right )+c_2 \left (1331-108 e^{10 t/3}\right )\right )}{42592} \\ \text {y3}(t)\to \frac {e^{5 t} \left (c_1 \left (720 e^{10 t/3}+1331\right )-c_2 \left (360 e^{10 t/3}+1331\right )\right )}{85184} \\ \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 51

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) - 2*y__2(t) + 2*y__3(t) + Derivative(y__1(t), t),0),Eq(2*y__1(t) - 7*y__2(t) + 2*y__3(t) + Derivative(y__2(t), t),0),Eq(10*y__1(t) - 10*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)
 

\[ \left [ y^{1}{\left (t \right )} = \frac {C_{1} e^{- 5 t}}{5} - \left (C_{2} - C_{3}\right ) e^{5 t}, \ y^{2}{\left (t \right )} = \frac {C_{1} e^{- 5 t}}{5} + C_{3} e^{5 t}, \ y^{3}{\left (t \right )} = C_{1} e^{- 5 t} + C_{2} e^{5 t}\right ] \]