67.7.8 problem Problem 4(d)

Internal problem ID [14048]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 4(d)
Date solved : Monday, March 31, 2025 at 08:22:58 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-2 y \left (t \right )+3 z \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right )+2 z \left (t \right )+2 \,{\mathrm e}^{-t}\\ \frac {d}{d t}z \left (t \right )&=-2 x \left (t \right )+2 y \left (t \right )-2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.183 (sec). Leaf size: 89

ode:=[diff(x(t),t) = 3*x(t)-2*y(t)+3*z(t), diff(y(t),t) = x(t)-y(t)+2*z(t)+2*exp(-t), diff(z(t),t) = -2*x(t)+2*y(t)-2*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 2 \,{\mathrm e}^{-t}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-2 t}+c_3 \,{\mathrm e}^{t} t \\ y \left (t \right ) &= {\mathrm e}^{-t}-\frac {c_1 \,{\mathrm e}^{t}}{2}+c_2 \,{\mathrm e}^{-2 t}-\frac {c_3 \,{\mathrm e}^{t} t}{2}+\frac {7 c_3 \,{\mathrm e}^{t}}{4} \\ z \left (t \right ) &= -2 \,{\mathrm e}^{-t}-c_1 \,{\mathrm e}^{t}-c_2 \,{\mathrm e}^{-2 t}-c_3 \,{\mathrm e}^{t} t +\frac {3 c_3 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 174

ode={D[x[t],t]==3*x[t]-2*y[t]+3*z[t],D[y[t],t]==x[t]-y[t]+2*z[t]+2*Exp[-t],D[z[t],t]==-2*x[t]+2*y[t]-2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{9} e^{-2 t} \left (18 e^t+e^{3 t} (c_1 (6 t+13)+c_3 (6 t+7)-6 c_2)-4 c_1+6 c_2-7 c_3\right ) \\ y(t)\to \frac {1}{9} e^{-2 t} \left (9 e^t+e^{3 t} (c_1 (4-3 t)+c_3 (7-3 t)+3 c_2)-4 c_1+6 c_2-7 c_3\right ) \\ z(t)\to \frac {1}{9} e^{-2 t} \left (-18 e^t+2 e^{3 t} (-(c_1 (3 t+2))-3 c_3 t+3 c_2+c_3)+4 c_1-6 c_2+7 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.265 (sec). Leaf size: 87

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*x(t) + 2*y(t) - 3*z(t) + Derivative(x(t), t),0),Eq(-x(t) + y(t) - 2*z(t) + Derivative(y(t), t) - 2*exp(-t),0),Eq(2*x(t) - 2*y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

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