12.22.12 problem section 10.5, problem 12

Internal problem ID [2265]
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 12
Date solved : Tuesday, March 04, 2025 at 01:52:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 85

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 127

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

\begin{align*} \text {y1}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (c_1 (4 t+2)-c_2 (4 t+1)+c_3)+c_2-c_3\right ) \\ \text {y2}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (4 (c_1-c_2) t+c_2+c_3)+c_2-c_3\right ) \\ \text {y3}(t)\to \frac {1}{2} e^{-2 t} \left (e^{6 t} (4 (c_1-c_2) t+c_2+c_3)-c_2+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.172 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-6*y__1(t) + 5*y__2(t) - 3*y__3(t) + Derivative(y__1(t), t),0),Eq(-2*y__1(t) + y__2(t) - 3*y__3(t) + Derivative(y__2(t), t),0),Eq(-2*y__1(t) - y__2(t) - y__3(t) + Derivative(y__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[y__1(t),y__2(t),y__3(t)],ics=ics)
 

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