12.21.15 problem section 10.4, problem 15

Internal problem ID [2253]
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 15
Date solved : Saturday, March 29, 2025 at 11:51:56 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=3 y_{1} \left (t \right )+y_{2} \left (t \right )-y_{3} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=3 y_{1} \left (t \right )+5 y_{2} \left (t \right )+y_{3} \left (t \right )\\ \frac {d}{d t}y_{3} \left (t \right )&=-6 y_{1} \left (t \right )+2 y_{2} \left (t \right )+4 y_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.182 (sec). Leaf size: 52

ode:=[diff(y__1(t),t) = 3*y__1(t)+y__2(t)-y__3(t), diff(y__2(t),t) = 3*y__1(t)+5*y__2(t)+y__3(t), diff(y__3(t),t) = -6*y__1(t)+2*y__2(t)+4*y__3(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= c_2 +c_3 \,{\mathrm e}^{6 t} \\ y_{2} \left (t \right ) &= -c_2 -c_3 \,{\mathrm e}^{6 t}+{\mathrm e}^{6 t} c_1 \\ y_{3} \left (t \right ) &= -4 c_3 \,{\mathrm e}^{6 t}+2 c_2 +{\mathrm e}^{6 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 93

ode={D[ y1[t],t]==3*y1[t]+1*y2[t]-1*y3[t],D[ y2[t],t]==3*y1[t]+5*y2[t]+1*y3[t],D[ y1[t],t]==-6*y1[t]+2*y2[t]+4*y3[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)\to \frac {1}{625} \left (-\left (c_1 \left (e^{6 t}-500\right )\right )-c_2 \left (e^{6 t}+125\right )\right ) \\ \text {y2}(t)\to \frac {1}{625} \left (c_2 \left (125-4 e^{6 t}\right )-4 c_1 \left (e^{6 t}+125\right )\right ) \\ \text {y3}(t)\to \frac {1}{625} \left (-\left (c_1 \left (e^{6 t}-1000\right )\right )-c_2 \left (e^{6 t}+250\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.114 (sec). Leaf size: 39

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) - y__2(t) + y__3(t) + Derivative(y__1(t), t),0),Eq(-3*y__1(t) - 5*y__2(t) - y__3(t) + Derivative(y__2(t), t),0),Eq(6*y__1(t) - 2*y__2(t) - 4*y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

\[ \left [ y^{1}{\left (t \right )} = \frac {C_{3}}{2} - \left (\frac {C_{1}}{3} - \frac {C_{2}}{3}\right ) e^{6 t}, \ y^{2}{\left (t \right )} = C_{2} e^{6 t} - \frac {C_{3}}{2}, \ y^{3}{\left (t \right )} = C_{1} e^{6 t} + C_{3}\right ] \]