85.92.4 problem 1 (d)

Internal problem ID [23053]
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 (d)
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 )+3 x \left (t \right )-6 y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ y \left (0\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.068 (sec). Leaf size: 65

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

\begin{align*} x \left (t \right ) &= \sqrt {6}\, {\mathrm e}^{\left (-3+\sqrt {6}\right ) t}-\sqrt {6}\, {\mathrm e}^{-\left (3+\sqrt {6}\right ) t} \\ y \left (t \right ) &= \frac {\sqrt {6}\, \left (\sqrt {6}\, {\mathrm e}^{\left (-3+\sqrt {6}\right ) t}+\sqrt {6}\, {\mathrm e}^{-\left (3+\sqrt {6}\right ) t}\right )}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 63

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

\begin{align*} x(t)&\to \sqrt {6} e^{-\left (\left (3+\sqrt {6}\right ) t\right )} \left (e^{2 \sqrt {6} t}-1\right )\\ y(t)&\to e^{-\left (\left (3+\sqrt {6}\right ) t\right )} \left (e^{2 \sqrt {6} t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.123 (sec). Leaf size: 58

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

\[ \left [ x{\left (t \right )} = - \sqrt {6} e^{- t \left (\sqrt {6} + 3\right )} + \sqrt {6} e^{- t \left (3 - \sqrt {6}\right )}, \ y{\left (t \right )} = e^{- t \left (\sqrt {6} + 3\right )} + e^{- t \left (3 - \sqrt {6}\right )}\right ] \]