Dy/dt=8x-t use the Laplace transform to solve the given system of differential equations

laplace-transform

#1

dy/dt=8x-t use the Laplace transform to solve the given system of differential equations

Answer:

I don’t know if it’s possible to solve this without knowing what x(0) or x’(0) are, because taking the Laplace transform will create both of those terms, and without knowing what they are we can’t eliminate them and come up with an answer (which you will see soon). Notice that we can solve each of these problems separately. A neat trick to simplify this equation is to take another derivative with respect to t of both sides. That turns the first equation (dx/dt=2y+e^t) into something that we can sub the second equation into. Check it out: d^2x/dt^2 = 2 dy/dt + e^t <— because the derivative of e^t = e^t Notice that we have a dy/dt in here and can now sub in our second equation. x" (I’ve turned the d^2x/dt^2 into x" here to make it easier to write) = 2(8x-t)+e^t. This simplifies to x"-16x=e^t-2t. Now do the Laplace transform of all the terms: s^2X(s)-sx(0)-x’(0)-16X(s)=(1/s)-(2/s^2). This is where we need to know what x(0) and x’(0) are. If you can sub them in, do so. From here, it’s easy. Solve for X(s) and then do a reverse Laplace transform to get your final answer.