85.92.5 problem 1 (e)

Internal problem ID [23054]
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 (e)
Date solved : Thursday, October 02, 2025 at 09:18:22 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+8 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 34

ode:=[diff(x(t),t) = x(t)+8*y(t), diff(y(t),t) = -2*x(t)-7*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_2 t +c_1 \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-3 t} \left (4 c_2 t +4 c_1 -c_2 \right )}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 149

ode={D[x[t],{t,1}]==x[t]-8*y[t], D[y[t],{t,1}]==-2*x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-\left (\left (3+4 \sqrt {2}\right ) t\right )} \left (c_1 \left (\left (2+\sqrt {2}\right ) e^{8 \sqrt {2} t}+2-\sqrt {2}\right )-2 \sqrt {2} c_2 \left (e^{8 \sqrt {2} t}-1\right )\right )\\ y(t)&\to \frac {1}{8} e^{-\left (\left (3+4 \sqrt {2}\right ) t\right )} \left (-\sqrt {2} c_1 \left (e^{8 \sqrt {2} t}-1\right )-2 c_2 \left (\left (\sqrt {2}-2\right ) e^{8 \sqrt {2} t}-2-\sqrt {2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 8*y(t) + Derivative(x(t), t),0),Eq(2*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 2 C_{1} \left (1 - \sqrt {2}\right ) e^{- t \left (3 + 4 \sqrt {2}\right )} - 2 C_{2} \left (1 + \sqrt {2}\right ) e^{- t \left (3 - 4 \sqrt {2}\right )}, \ y{\left (t \right )} = C_{1} e^{- t \left (3 + 4 \sqrt {2}\right )} + C_{2} e^{- t \left (3 - 4 \sqrt {2}\right )}\right ] \]