81.17.6 problem 21-55

Internal problem ID [21739]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 21. Applications of second order differential equations
Problem number : 21-55
Date solved : Thursday, October 02, 2025 at 08:01:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )&=2 y \left (t \right )-5 z \left (t \right )\\ \frac {d}{d t}z \left (t \right )&=4 y \left (t \right )-2 z \left (t \right ) \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 49

ode:=[diff(y(t),t) = 2*y(t)-5*z(t), diff(z(t),t) = 4*y(t)-2*z(t)]; 
dsolve(ode);
 

\begin{align*} y \left (t \right ) &= c_1 \sin \left (4 t \right )+c_2 \cos \left (4 t \right ) \\ z \left (t \right ) &= -\frac {4 c_1 \cos \left (4 t \right )}{5}+\frac {4 c_2 \sin \left (4 t \right )}{5}+\frac {2 c_1 \sin \left (4 t \right )}{5}+\frac {2 c_2 \cos \left (4 t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 58

ode={D[y[t],t]==2*y[t]-5*z[t], D[z[t],t]==4*y[t]-2*z[t]}; 
ic={}; 
DSolve[{ode,ic},{z[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to c_1 \cos (4 t)+\frac {1}{4} (2 c_1-5 c_2) \sin (4 t)\\ z(t)&\to c_2 \cos (4 t)+\frac {1}{2} (2 c_1-c_2) \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-2*y(t) + 5*z(t) + Derivative(y(t), t),0),Eq(-4*y(t) + 2*z(t) + Derivative(z(t), t),0)] 
ics = {} 
dsolve(ode,func=[y(t),z(t)],ics=ics)
 

\[ \left [ y{\left (t \right )} = \left (\frac {C_{1}}{2} - C_{2}\right ) \cos {\left (4 t \right )} - \left (C_{1} + \frac {C_{2}}{2}\right ) \sin {\left (4 t \right )}, \ z{\left (t \right )} = C_{1} \cos {\left (4 t \right )} - C_{2} \sin {\left (4 t \right )}\right ] \]