83.13.4 problem B(5)

Internal problem ID [22012]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IX. System of equations. Ex. XVII at page 154
Problem number : B(5)
Date solved : Thursday, October 02, 2025 at 08:21:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-2 x \left (t \right )+\frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=1\\ \frac {d}{d t}y \left (t \right )+\frac {d}{d t}z \left (t \right )+z \left (t \right )&=2\\ 3 x \left (t \right )+\frac {d}{d t}z \left (t \right )+z \left (t \right )&=3 \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: 58

ode:=[diff(x(t),t)-2*x(t)+diff(y(t),t)-2*y(t) = 1, diff(y(t),t)+diff(z(t),t)+z(t) = 2, 3*x(t)+diff(z(t),t)+z(t) = 3]; 
ic:=[x(0) = 0, y(0) = 0, z(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{5}-\frac {8 \,{\mathrm e}^{-3 t}}{15}+\frac {1}{3} \\ y \left (t \right ) &= \frac {3 \,{\mathrm e}^{2 t}}{10}+\frac {8 \,{\mathrm e}^{-3 t}}{15}-\frac {5}{6} \\ z \left (t \right ) &= -\frac {{\mathrm e}^{2 t}}{5}+2-{\mathrm e}^{-t}-\frac {4 \,{\mathrm e}^{-3 t}}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 78

ode={D[x[t],t]-2*x[t]+D[y[t],t]-2*y[t]==1,D[y[t],t]+D[z[t],t]+z[t]==2,3*x[t]+D[z[t],t]+z[t]==3}; 
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 \frac {1}{15} \left (-8 e^{-3 t}+3 e^{2 t}+5\right )\\ y(t)&\to \frac {1}{30} \left (16 e^{-3 t}+9 e^{2 t}-25\right )\\ z(t)&\to -\frac {4 e^{-3 t}}{5}-e^{-t}-\frac {e^{2 t}}{5}+2 \end{align*}

✓ Sympy. Time used: 0.234 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*x(t) - 2*y(t) + Derivative(x(t), t) + Derivative(y(t), t) - 1,0),Eq(z(t) + Derivative(y(t), t) + Derivative(z(t), t) - 2,0),Eq(3*x(t) + z(t) + Derivative(z(t), t) - 3,0)] 
ics = {x(0): 0, y(0): 0, z(0): 0} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {e^{2 t}}{5} + \frac {1}{3} - \frac {8 e^{- 3 t}}{15}, \ y{\left (t \right )} = \frac {3 e^{2 t}}{10} - \frac {5}{6} + \frac {8 e^{- 3 t}}{15}, \ z{\left (t \right )} = - \frac {e^{2 t}}{5} + 2 - e^{- t} - \frac {4 e^{- 3 t}}{5}\right ] \]