83.13.2 problem A(2)

Internal problem ID [22010]
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 : A(2)
Date solved : Thursday, October 02, 2025 at 08:21:38 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )-3 z \left (t \right )&=5\\ y \left (t \right )-\frac {d}{d t}z \left (t \right )-x \left (t \right )&=3-2 t\\ z \left (t \right )+\frac {d}{d t}x \left (t \right )&=-1 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 61

ode:=[diff(y(t),t)-3*z(t) = 5, y(t)-diff(z(t),t)-x(t) = 3-2*t, z(t)+diff(x(t),t) = -1]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 165

ode={D[y[t],t]-3*z[t]==5,y[t]-D[z[t],t]-x[t]==3-2*t,z[t]+D[x[t],t]==-1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{8} e^{-2 t} \left (2 e^{2 t} (4 t-3+3 c_1+c_2)+(c_1-c_2-2 c_3) e^{4 t}+c_1-c_2+2 c_3\right )\\ y(t)&\to \frac {1}{8} \left (-8 t-3 (c_1-c_2+2 c_3) e^{-2 t}+(-3 c_1+3 c_2+6 c_3) e^{2 t}+2 (9+3 c_1+c_2)\right )\\ z(t)&\to \frac {1}{4} e^{-2 t} \left (-8 e^{2 t}+(-c_1+c_2+2 c_3) e^{4 t}+c_1-c_2+2 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 71

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-3*z(t) + Derivative(y(t), t) - 5,0),Eq(2*t - x(t) + y(t) - Derivative(z(t), t) - 3,0),Eq(z(t) + Derivative(x(t), t) + 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} + \frac {C_{2} e^{- 2 t}}{2} - \frac {C_{3} e^{2 t}}{2} + t - \frac {3}{4}, \ y{\left (t \right )} = C_{1} - \frac {3 C_{2} e^{- 2 t}}{2} + \frac {3 C_{3} e^{2 t}}{2} - t + \frac {9}{4}, \ z{\left (t \right )} = C_{2} e^{- 2 t} + C_{3} e^{2 t} - 2\right ] \]