12.22.11 problem section 10.5, problem 11

Internal problem ID [2264]
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 11
Date solved : Tuesday, March 04, 2025 at 01:52:40 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 68

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

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 126

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

\begin{align*} \text {y1}(t)\to e^{2 t} \left ((c_1-c_2-c_3) e^{2 t}+c_2+c_3\right ) \\ \text {y2}(t)\to \frac {1}{2} \left (3 (c_2+c_3) e^{2 t}-e^{4 t} (4 (c_1-c_2-c_3) t+c_2+3 c_3)\right ) \\ \text {y3}(t)\to -\frac {1}{2} e^{2 t} \left (-e^{2 t} (4 (c_1-c_2-c_3) t+c_2+3 c_3)+c_2+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.156 (sec). Leaf size: 75

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

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