Use the Laplace transform to solve the given system of differential equations. d2x dt2 + x − y = 0 d2y dt2 + y − x = 0 x(0) = 0, x'(0) = −6, y(0) = 0, y'(0) = 1

I'm going to assume that "−" is supposed to be some kind of minus character, so that the given system of DEs is supposed to be

Take the Laplace transform of both sides of both equations. Recall the transform for a second-order derivative,

where F(s) denotes the transform of f(t). You end up with

and solving for X(s) and Y(s) (nothing tricky here, just two linear equations) gives

Now solve for x(t) and y(t) by computing the inverse transforms. To start, split up both X(s) and Y(s) into partial fractions.

• Solving for x(t) :

Taking the inverse transform, you get

• Solving for y(t) :

Inverse transform:

"image"

step-by-step explanation:

in most of these problems, the notation is that the point with a "prime" is the image of the point without. that is, b' is the image of b.

here, there is no dilation or reflection. there is simply translation (with no direction shown). which figure you call the "preimage" and which one you call the "image" is up to you. as a matter of convention, the naming is as described above.