58.19.1 problem 1

Internal problem ID [14923]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.7. Exercises page 375
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:56:36 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.114 (sec). Leaf size: 64

ode:=[diff(x(t),t) = x(t)+y(t)-z(t), diff(y(t),t) = 2*x(t)+3*y(t)-4*z(t), diff(z(t),t) = 4*x(t)+y(t)-4*z(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{2 t} \\ y \left (t \right ) &= 7 c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t}+2 c_3 \,{\mathrm e}^{2 t} \\ z \left (t \right ) &= 11 c_1 \,{\mathrm e}^{-3 t}+c_2 \,{\mathrm e}^{t}+c_3 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 198

ode={D[x[t],t]==x[t]+y[t]-z[t],D[y[t],t]==2*x[t]+3*y[t]-4*z[t],D[z[t],t]==4*x[t]+y[t]-4*z[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-4 e^{5 t}-1\right )+2 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+c_3\right )\\ y(t)&\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-8 e^{5 t}-7\right )+4 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+7 c_3\right )\\ z(t)&\to \frac {1}{10} e^{-3 t} \left (c_1 \left (15 e^{4 t}-4 e^{5 t}-11\right )+2 (5 c_2-3 c_3) e^{5 t}+5 (c_3-2 c_2) e^{4 t}+11 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) - y(t) + z(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 3*y(t) + 4*z(t) + Derivative(y(t), t),0),Eq(-4*x(t) - y(t) + 4*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{- 3 t}}{11} + C_{2} e^{t} + C_{3} e^{2 t}, \ y{\left (t \right )} = \frac {7 C_{1} e^{- 3 t}}{11} + C_{2} e^{t} + 2 C_{3} e^{2 t}, \ z{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{t} + C_{3} e^{2 t}\right ] \]