78.5.19 problem 7.g

Internal problem ID [21058]
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.g
Date solved : Thursday, October 02, 2025 at 07:01:51 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=8 x \left (t \right )-5 y\\ y^{\prime }&=16 x \left (t \right )+8 y \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.187 (sec). Leaf size: 63

ode:=[diff(x(t),t) = 8*x(t)-5*y(t), diff(y(t),t) = 16*x(t)+8*y(t)]; 
ic:=[x(0) = 1, y(0) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{8 t} \left (\cos \left (4 \sqrt {5}\, t \right )+\frac {\sqrt {5}\, \sin \left (4 \sqrt {5}\, t \right )}{4}\right ) \\ y \left (t \right ) &= -\frac {4 \,{\mathrm e}^{8 t} \sqrt {5}\, \left (\frac {\sqrt {5}\, \cos \left (4 \sqrt {5}\, t \right )}{4}-\sin \left (4 \sqrt {5}\, t \right )\right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 81

ode={D[x[t],t]==8*x[t]-5*y[t],D[y[t],t]==16*x[t]+8*y[t]}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{8 t} \left (\sqrt {5} \sin \left (4 \sqrt {5} t\right )+4 \cos \left (4 \sqrt {5} t\right )\right )\\ y(t)&\to \frac {1}{5} e^{8 t} \left (4 \sqrt {5} \sin \left (4 \sqrt {5} t\right )-5 \cos \left (4 \sqrt {5} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 80

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-8*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-16*x(t) - 8*y(t) + Derivative(y(t), t),0)] 
ics = {x(0): 1, y(0): -1} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {\sqrt {5} e^{8 t} \sin {\left (4 \sqrt {5} t \right )}}{4} + e^{8 t} \cos {\left (4 \sqrt {5} t \right )}, \ y{\left (t \right )} = \frac {4 \sqrt {5} e^{8 t} \sin {\left (4 \sqrt {5} t \right )}}{5} - e^{8 t} \cos {\left (4 \sqrt {5} t \right )}\right ] \]