85.92.6 problem 1 (f)

Internal problem ID [23055]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 11. Matrix eigenvalue methods for systems of linear differential equations. A Exercises at page 528
Problem number : 1 (f)
Date solved : Thursday, October 02, 2025 at 09:18:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-12 x \left (t \right )-7 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=19 x \left (t \right )+11 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 83

ode:=[diff(x(t),t) = -12*x(t)-7*y(t), diff(y(t),t) = 19*x(t)+11*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_2 -\cos \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_1 -23 \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_1 -23 \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_2 \right )}{14} \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 115

ode={D[x[t],{t,1}]==-12*x[t]-7*y[t], D[y[t],{t,1}]==19*x[t]+11*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{3} e^{-t/2} \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} (23 c_1+14 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right )\\ y(t)&\to \frac {1}{3} e^{-t/2} \left (3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (38 c_1+23 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 97

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(12*x(t) + 7*y(t) + Derivative(x(t), t),0),Eq(-19*x(t) - 11*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (\frac {23 C_{1}}{38} + \frac {\sqrt {3} C_{2}}{38}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - \left (\frac {\sqrt {3} C_{1}}{38} - \frac {23 C_{2}}{38}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {3} t}{2} \right )} - C_{2} e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {3} t}{2} \right )}\right ] \]