85.88.9 problem 1 (i)

Internal problem ID [23036]
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 (i)
Date solved : Thursday, October 02, 2025 at 09:17:40 PM
CAS classification : system_of_ODEs

\begin{align*} 2 x \left (t \right )-\frac {d}{d t}y \left (t \right )-5 y \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 y \left (t \right )&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=-10 \\ \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 25

ode:=[2*x(t)-diff(y(t),t)-5*y(t) = 0, diff(x(t),t)+x(t)+2*y(t) = 0]; 
ic:=[x(0) = 0, y(0) = -10]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 20 \,{\mathrm e}^{-3 t} t \\ y \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (40 t -20\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 28

ode={2*x[t]-D[y[t],t]-5*y[t]==0, D[x[t],t]+x[t]+ 2*y[t]==0}; 
ic={x[0]==0,y[0]==-10}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 20 e^{-3 t} t\\ y(t)&\to 10 e^{-3 t} (2 t-1) \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) - 5*y(t) - Derivative(y(t), t),0),Eq(x(t) + 2*y(t) + Derivative(x(t), t),0)] 
ics = {x(0): 0, y(0): -10} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 20 t e^{- 3 t}, \ y{\left (t \right )} = 20 t e^{- 3 t} - 10 e^{- 3 t}\right ] \]