5 Laplace Transforms with Solutions

Photo by Towfiqu barbhuiya on Unsplash

A Guide to how to evaluate different Laplace Transforms.

For each of the problems set out below there is a question which I very much suggest you try to answer before looking at the answer at the end. If you come across a question you can’t solve then look up the answer at the bottom and try to generate your own problems that test your understanding of what you have learnt. This is the best way to learn.

Questions

Powers

Powers of t (made with Latex)

Trigonometry

Trigonometry (made with Latex)

Piecewise

A Piecewise Function (made with Latex)

Shifted

Exponential Trigonometry (made with Latex)

Changes?

t times the sine (made with Latex)

Solutions

Powers

For this Laplace transform we can use the simple power rule which states.

power rule (made with Latex)

Then since we can separate added terms together, we can evaluate the whole expression.

solving our Laplace transform (made with Latex)

And that in this case is the most simplified form.

Trigonometry

For this expression we need to remember two more well known Laplace transforms

transform of sine (made with Latex)

transform of cosine (made with Latex)

Using these and that nice property of splitting a Laplace transform up we can evaluate the whole expression.

Solution of our Problem (made with Latex)

And there we have evaluated a trigonometric Laplace transform.

Piecewise

For any piecewise function we need to write it first in terms of something called the unit step function. I have made another article on exactly how to do this if you don’t know how to. But in short it comes down to using the unit step function as a kind of on/off switch for the different parts of the function.

Here I have turned the statement above into this form.

Converted into terms of the unit step function (made with Latex)

Then from this we need one more nice property of the Laplace transform.

step function property (made with Latex)

Using this we can then solve the Laplace transform.

solution to our problem (made with Latex)

And there we have it. We could expand out, the first term if we were solving this as an ODE but, for making it look nice, this is the nicest way.

Shifted

For this one, as said in its title, we use the Laplace transforms shifting property. This means that,

shifting rule (made with Latex)

Here I have used a capital F to denote the Laplace transform of the function f, this is a standard notation that you might come across, when dealing with Laplace transforms.

We can use this one property to solve the whole expression.

solution to our problem (made with Latex)

We could simplify this further as well, but this seems like quite a nice form to have it in.

Changes?

Now this title was a big clue with what to do to solve it. We have one not as well-known property of the Laplace transform that is extremely useful, that:

derivative rule for this transform (made with Latex)

And again, it is not too hard once we have this property.

solution to our problem (made with Latex)

And there we have the solution.

As an extra challenge if you want it, I encourage you to try deriving all the formulas we used today. And as always,

Have fun and never stop solving.