Diff. Eq. - Solve the following equation using Laplace transforms

Exmpl. #1) Solve the following equation using Laplace transforms. Where y(0) = 0, y'(0) = 0 and x(t) is a step change of magnitude 2.

Standard procedure of these problems, taking the Laplace transform of both sides!

But let's start with the left side first:

(*Laplace transforms of derivatives must be known before calculating... click here!)

After plugging in f(0) and f'(0), and then simplifying, we get:

Now let's do the Laplace transform of the right side of the equation. Remember that it was given to us that x(t) is a step change of magnitude 2. Laplace transform of a step function is always 1/s, and we will multiply it by 2 to represent the change of magnitude.

Time to factor out Y, which represents the Laplace transform of a function y(t).

---> (*Apply a/b/c/1 fraction rule!)

Now, we will use partial fraction decomposition to solve for Y.

Let's set this up.

Now distribute and simplify.

Group and equate the terms by power (right and left side of the equal sign)

You will find that,

A = 1/3

B = -1

C =
2/3

Now we take the inverse laplace transform of the equation to get our answer!

A standard laplace transform table, should help you easily calculate these values

*Inv. Laplace of variable is the same variable, but in the time domain. So, Inverse Laplace of Y is y(t)

and here is our answer: