85.90.1 problem 1

Internal problem ID [23046]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. C Exercises at page 500
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:17:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-5 x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 z \left (t \right )&=24 \,{\mathrm e}^{-t}\\ \frac {d}{d t}x \left (t \right )-x \left (t \right )-y \left (t \right )&=0\\ 5 \frac {d}{d t}y \left (t \right )-11 y \left (t \right )+2 \frac {d}{d t}z \left (t \right )-2 z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 79

ode:=[diff(x(t),t)-5*x(t)+diff(y(t),t)+2*z(t) = 24*exp(-t), diff(x(t),t)-x(t)-y(t) = 0, 5*diff(y(t),t)-11*y(t)+2*diff(z(t),t)-2*z(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{3 t} c_2}{2}+{\mathrm e}^{2 t} c_3 +\frac {{\mathrm e}^{t} c_1}{2}+2 \,{\mathrm e}^{-t} \\ y \left (t \right ) &= {\mathrm e}^{2 t} c_3 +{\mathrm e}^{3 t} c_2 -4 \,{\mathrm e}^{-t} \\ z \left (t \right ) &= -{\mathrm e}^{3 t} c_2 +\frac {{\mathrm e}^{2 t} c_3}{2}+{\mathrm e}^{t} c_1 +16 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 186

ode={D[x[t],{t,1}]-5*x[t]+D[y[t],t]+2*z[t]==24*Exp[-t], D[x[t],{t,1}]-x[t]-y[t]==0,5*D[y[t],t]-11*y[t]+2*D[z[t],t]-2*z[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{2} e^{-t} \left ((6 c_1-5 c_2-2 c_3) e^{2 t}+(4 c_1-3 c_2-2 c_3) e^{4 t}+(-8 c_1+8 c_2+4 c_3) e^{3 t}+4\right )\\ y(t)&\to e^{-t} \left ((4 c_1-3 c_2-2 c_3) e^{4 t}+(-4 c_1+4 c_2+2 c_3) e^{3 t}-4\right )\\ z(t)&\to 16 e^{-t}+(6 c_1-5 c_2-2 c_3) e^t+(-2 c_1+2 c_2+c_3) e^{2 t}+(-4 c_1+3 c_2+2 c_3) e^{3 t} \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-5*x(t) + 2*z(t) + Derivative(x(t), t) + Derivative(y(t), t) - 24*exp(-t),0),Eq(-x(t) - y(t) + Derivative(x(t), t),0),Eq(-11*y(t) - 2*z(t) + 5*Derivative(y(t), t) + 2*Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} e^{2 t} + \frac {C_{2} e^{t}}{2} - \frac {C_{3} e^{3 t}}{2} + 2 e^{- t}, \ y{\left (t \right )} = 2 C_{1} e^{2 t} - C_{3} e^{3 t} - 4 e^{- t}, \ z{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{t} + C_{3} e^{3 t} + 16 e^{- t}\right ] \]