85.88.10 problem 1 (j)

Internal problem ID [23037]
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 (j)
Date solved : Thursday, October 02, 2025 at 09:17:41 PM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )-6 x \left (t \right )+3 \frac {d}{d t}y \left (t \right )-2 y \left (t \right )&=0\\ 7 \frac {d}{d t}x \left (t \right )+4 x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+20 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 49

ode:=[2*diff(x(t),t)-6*x(t)+3*diff(y(t),t)-2*y(t) = 0, 7*diff(x(t),t)+4*x(t)+7*diff(y(t),t)+20*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (4 t \right )+c_2 \cos \left (4 t \right ) \\ y \left (t \right ) &= -\frac {14 c_1 \cos \left (4 t \right )}{37}+\frac {14 c_2 \sin \left (4 t \right )}{37}-\frac {27 c_1 \sin \left (4 t \right )}{37}-\frac {27 c_2 \cos \left (4 t \right )}{37} \\ \end{align*}

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

ode={2*D[x[t],{t,1}]-6*x[t]+3*D[y[t],t]-2*y[t]==0, 7*D[x[t],t]+4*x[t]+ 7*D[y[t],{t,1}]+20*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 \cos (4 t)-\frac {1}{14} (27 c_1+37 c_2) \sin (4 t)\\ y(t)&\to c_2 \cos (4 t)+\frac {1}{14} (25 c_1+27 c_2) \sin (4 t) \end{align*}

✓ Sympy. Time used: 0.082 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-6*x(t) - 2*y(t) + 2*Derivative(x(t), t) + 3*Derivative(y(t), t),0),Eq(4*x(t) + 20*y(t) + 7*Derivative(x(t), t) + 7*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {14 C_{1}}{25} - \frac {27 C_{2}}{25}\right ) \sin {\left (4 t \right )} - \left (\frac {27 C_{1}}{25} + \frac {14 C_{2}}{25}\right ) \cos {\left (4 t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (4 t \right )} - C_{2} \sin {\left (4 t \right )}\right ] \]