85.88.7 problem 1 (g)

Internal problem ID [23034]
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 (g)
Date solved : Thursday, October 02, 2025 at 09:17:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+3 y \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )-2 x \left (t \right )+5 \frac {d}{d t}y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.079 (sec). Leaf size: 47

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 55

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

\begin{align*} x(t)&\to e^{-t} (c_1 \cos (t)-(2 c_1+5 c_2) \sin (t))\\ y(t)&\to e^{-t} (c_2 \cos (t)+(c_1+2 c_2) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 46

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

\[ \left [ x{\left (t \right )} = - \left (C_{1} - 2 C_{2}\right ) e^{- t} \sin {\left (t \right )} - \left (2 C_{1} + C_{2}\right ) e^{- t} \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} e^{- t} \cos {\left (t \right )} - C_{2} e^{- t} \sin {\left (t \right )}\right ] \]