83.15.16 problem 6 (a)

Internal problem ID [22040]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 6 (a)
Date solved : Thursday, October 02, 2025 at 08:22:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )-y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+2 y \left (t \right )+\frac {d}{d t}z \left (t \right )+2 z \left (t \right )&=2\\ x \left (t \right )+\frac {d}{d t}z \left (t \right )-z \left (t \right )&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.063 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{-2 t}-4 \,{\mathrm e}^{-t}+1 \\ y \left (t \right ) &= 2 \,{\mathrm e}^{-t}-2 \,{\mathrm e}^{-2 t} \\ z \left (t \right ) &= 1+{\mathrm e}^{-2 t}-2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 52

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

\begin{align*} x(t)&\to 3 e^{-2 t}-4 e^{-t}+1\\ y(t)&\to 2 e^{-2 t} \left (e^t-1\right )\\ z(t)&\to e^{-2 t} \left (e^t-1\right )^2 \end{align*}

✓ Sympy. Time used: 0.199 (sec). Leaf size: 61

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

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