12.22.27 problem section 10.5, problem 27

Internal problem ID [2280]
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 27
Date solved : Tuesday, March 04, 2025 at 01:52:57 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.051 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 108

ode={D[ y1[t],t]==0*y1[t]+2*y2[t]-2*y3[t],D[ y2[t],t]==-1*y1[t]+5*y2[t]-3*y3[t],D[ y3[t],t]==1*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 -e^{2 t} (c_1 (2 t-1)+2 (c_3-c_2) t) \\ \text {y2}(t)\to e^{2 t} \left (-2 (c_1-c_2+c_3) t^2-(c_1-3 c_2+3 c_3) t+c_2\right ) \\ \text {y3}(t)\to e^{2 t} \left (-2 (c_1-c_2+c_3) t^2+(c_1+c_2-c_3) t+c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.179 (sec). Leaf size: 97

from sympy import * 
t = symbols("t") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-2*y__2(t) + 2*y__3(t) + Derivative(y__1(t), t),0),Eq(y__1(t) - 5*y__2(t) + 3*y__3(t) + Derivative(y__2(t), t),0),Eq(-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)
 

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