Consider the system of differential equations
x' = 5x + 3y,
y' = 3x + 5y.
Verify that for any constants $c_1$ and $c_2$, the functions
x(t) = $c_1e^{2t} + c_2e^{8t}$,
y(t) = $-c_1e^{2t} + c_2e^{8t}$,
satisfy the system of differential equations by plugging the guess functions into each side of each equation. Enter $c_1$ as c1 and $c_2$ as c2.
a. Find an expression for each term in the equation x' = 5x + 3y in terms of the variable t. (Enter the terms in the order given.)
Notice that the two sides of the equation above are actually equal, showing that the guess functions work in this ODE.
b. Find an expression for each term in the equation y' = 3x + 5y in terms of the variable t. (Enter the terms in the order given.)
Notice that the two sides of the equation above are actually equal, showing that the guess functions work in this ODE.
c. Suppose the system of ODEs has initial conditions x(0) = 0 and y(0) = 6. Calculate the values of $c_1$ and $c_2$ that satisfy the IVP data.
$c_1$ = and $c_2$ =
Use the applet to plot a direction field for the system, including sample solution curves. The plot must be included in your submitted
work to receive full credit.
