82.7.8 problem 34-11

Internal problem ID [21864]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 34. Simulataneous linear differential equations. Page 1118
Problem number : 34-11
Date solved : Thursday, October 02, 2025 at 08:03:02 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )&=y \left (t \right )-3 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=2 y \left (t \right )-4 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 34

ode:=[diff(y(t),t) = y(t)-3*z(t), diff(z(t),t) = 2*y(t)-4*z(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 60

ode={D[y[t],t]==y[t]-3*z[t],D[z[t],t]==2*y[t]-4*z[t]}; 
ic={}; 
DSolve[{ode,ic},{z[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-2 t} \left (c_1 \left (3 e^t-2\right )-3 c_2 \left (e^t-1\right )\right )\\ z(t)&\to e^{-2 t} \left (2 c_1 \left (e^t-1\right )+c_2 \left (3-2 e^t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.048 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
z = Function("z") 
y = Function("y") 
ode=[Eq(-y(t) + 3*z(t) + Derivative(y(t), t),0),Eq(-2*y(t) + 4*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[z(t),y(t)],ics=ics)
 

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