85.87.1 problem 1

Internal problem ID [23025]
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. B Exercises at page 491
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:17:35 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=0 \\ z \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.078 (sec). Leaf size: 86

ode:=[diff(x(t),t) = x(t)+y(t), diff(y(t),t) = x(t)-y(t), diff(z(t),t) = 2*y(t)]; 
ic:=[x(0) = 2, y(0) = 0, z(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 101

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

\begin{align*} x(t)&\to \frac {1}{2} e^{-\sqrt {2} t} \left (\left (2+\sqrt {2}\right ) e^{2 \sqrt {2} t}+2-\sqrt {2}\right )\\ y(t)&\to \frac {e^{-\sqrt {2} t} \left (e^{2 \sqrt {2} t}-1\right )}{\sqrt {2}}\\ z(t)&\to e^{-\sqrt {2} t}+e^{\sqrt {2} t}-2 \end{align*}

✓ Sympy. Time used: 0.194 (sec). Leaf size: 90

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

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