85.88.3 problem 1 (c)

Internal problem ID [23030]
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 (c)
Date solved : Thursday, October 02, 2025 at 09:17:37 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+5 x \left (t \right )+3 \frac {d}{d t}y \left (t \right )-11 y \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )+3 x \left (t \right )+\frac {d}{d t}y \left (t \right )-7 y \left (t \right )&=0 \end{align*}

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

ode:=[diff(x(t),t)+5*x(t)+3*diff(y(t),t)-11*y(t) = 0, diff(x(t),t)+3*x(t)+diff(y(t),t)-7*y(t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 42

ode={D[x[t],{t,1}]+5*x[t]+3*D[y[t],t]-11*y[t]==0, D[x[t],{t,1}]+3*x[t]+ D[y[t],{t,1}]-7*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.066 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) - 11*y(t) + Derivative(x(t), t) + 3*Derivative(y(t), t),0),Eq(3*x(t) - 7*y(t) + Derivative(x(t), t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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