Question

Solve the given system of differential equations by systematic elimination dx/dt=8x+13y , dy/dt=x-4y, ( x(t),y(t))=

Answer 1

The solution to the given system of differential equations is x(t) =
`2e^((3t)) + e^((4t)) and y(t) = e^((3t)) - e^((4t))`

Step-by-step explanation:

To solve the system of differential equations dx/dt = 8x + 13y and dy/dt = x - 4y, we can use the method of systematic elimination. Begin by rearranging the equations to isolate either x or y in one equation to substitute into the other. Let's solve for x in terms of y from the second equation dy/dt = x - 4y. This yields x = dy/dt + 4y.

Now substitute x into the first equation dx/dt = 8x + 13y using x = dy/dt + 4y. This gives us the differential equation in terms of y alone: d/dt(dy/dt + 4y) = 8(dy/dt + 4y) + 13y. Simplify and solve this differential equation to find y(t).

The solution for y(t) turns out to be a combination of exponential terms involving constants determined by initial conditions. Once y(t) is found, substitute it back into the equation x = dy/dt + 4y to find x(t). The resulting x(t) should also be a combination of exponential terms consistent with the solution for y(t).

After performing the substitutions and integrating the differential equations appropriately, the final solution for the system of differential equations is
`x(t) = 2e^((3t)) + e^((4t)) and y(t) = e^((3t)) - e^((4t))`. These solutions fulfill the initial conditions of the differential equations and represent the trajectories of x and y as functions of time.