stumbled upon it years ago but I’m terrible at math so kinda forgot about it for years, recently stumbled upon it again so out of pure curiosity, what is it? does it have real world use or is it something purely theoretical?
In: Mathematics
So the topic is a bit iffy for eli5 but can eli16
The basis of this is that you understand what an algebraic expression is. Like 4x = 20 you know x must be 5.
In calculus we care a lot about the slopes of functions. You’re probably familiar with something like y = 3x + 2 and it has a slope of 3 everywhere. A big part of calculus is finding the slopes of any functions at any point. The line example y = mx + b is taught in algebra because the slope is the same everywhere. But what about the slope of x^4 + 3x + 2?? Calculus can answer this.
Turns out slopes are useful for physics/science. If you’ve ever taken physics you spend a lot of time doing problems with newtons second law F = ma and they have you solving for force or acceleration all the time. People in the real world don’t care about acceleration all that much but what we do care about is the positions of things. We know that acceleration is the change in velocity over time. Or in other words, the slope of the velocity function. What’s velocity? It’s change in displacement over time (position with a direction). So most of the time if we want to know where a particle is located we can use high school physics to set up acceleration = something. Then we have to find a function where the slope of the slope is equal to that something. If that sounds hard, most of the time it is. But here’s where the Laplace transform comes in. If you take the Laplace transform or that equation, instead of dealing with slopes and calculus it turns the equation into an algebraic expression where you can use your seventh grade algebra skills to solve for a variable. Then you can do the inverse transformation to get back what you were looking for. The Laplace transformation and inverse transformations are things you rarely “do” but look up in a table due to patterns.
The tl;dr answer Is that the Laplace transform turns a differential equation into an algebraic expression.
The eli5 that glosses over a lot is it turns an equation you need to be a sophomore in college to solve into one you need to be a seventh grader to solve. It doesn’t always work nicely but it’s a cool trick to try when faced with a complicated problem.
If you’re interested in learning more, search “what is a differential equation” because it’s mainly a technique used to solve those.
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