85.88.8 problem 1 (h)

Internal problem ID [23035]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 499
Problem number : 1 (h)
Date solved : Thursday, October 02, 2025 at 09:17:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )-y \left (t \right )&=0\\ 5 x \left (t \right )+\frac {d}{d t}y \left (t \right )-3 y \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 35

ode:=[diff(x(t),t)-x(t)-y(t) = 0, 5*x(t)+diff(y(t),t)-3*y(t) = 0]; 
ic:=[x(0) = 2, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (-\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) \\ y \left (t \right ) &= -5 \,{\mathrm e}^{2 t} \sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 38

ode={D[x[t],{t,1}]-x[t]-y[t]==0, 5*x[t]+ D[y[t],{t,1}]-3*y[t]==0}; 
ic={x[0]==2,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{2 t} (2 \cos (2 t)-\sin (2 t))\\ y(t)&\to -5 e^{2 t} \sin (2 t) \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 39

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

\[ \left [ x{\left (t \right )} = - e^{2 t} \sin {\left (2 t \right )} + 2 e^{2 t} \cos {\left (2 t \right )}, \ y{\left (t \right )} = - 5 e^{2 t} \sin {\left (2 t \right )}\right ] \]