78.5.16 problem 7.d

Internal problem ID [21055]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 7.d
Date solved : Thursday, October 02, 2025 at 07:01:49 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=4 x \left (t \right )-13 y\\ y^{\prime }&=2 x \left (t \right )-6 y \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.154 (sec). Leaf size: 29

ode:=[diff(x(t),t) = 4*x(t)-13*y(t), diff(y(t),t) = 2*x(t)-6*y(t)]; 
ic:=[x(0) = 2, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (10 \sin \left (t \right )+2 \cos \left (t \right )\right ) \\ y \left (t \right ) &= 4 \,{\mathrm e}^{-t} \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 31

ode={D[x[t],t]==4*x[t]-13*y[t],D[y[t],t]==2*x[t]-6*y[t]}; 
ic={x[0]==2,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to 2 e^{-t} (5 \sin (t)+\cos (t))\\ y(t)&\to 4 e^{-t} \sin (t) \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 13*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 6*y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 2, y(0): 0} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 10 e^{- t} \sin {\left (t \right )} + 2 e^{- t} \cos {\left (t \right )}, \ y{\left (t \right )} = 4 e^{- t} \sin {\left (t \right )}\right ] \]