# Laplace Transform A laplace transform is meant to convert equations that have derivatives in them, into an equation that is algebraic. This will be a new equation, with a new variable, involving *no* derivatives. An example of this would be: $\text{Differential Equation: } y'' - y' - 6y = 0$ $\downarrow \;\;\; \text{Laplace Transform} \;\;\; \downarrow$ $\text{Algebraic Equation: } (s^2 - s -6)Y(s) -2s + 3 = 0$ So, put simply, one of the main powers of the laplace transform is: > The **Laplace Transform** can convert differential equations in to algebraic equations that we are more easily able to solve. ### Notes * We see some similarities/common themes between Laplace Transforms and change of variables. * While the [Fourier Transform](Fourier%20Transform.md) tells us which frequencies or sinsusoids are present in a function, the laplace transform tells us which sinusoids *and* **exponentials** are present in a function. The Fourier transform can be thought of as a "slice" of the laplace transform. ### Equations  ### In Progress https://photos.google.com/photo/AF1QipN79gdw55HQy5ZYYL-FE2sH\_dpbMI0dN6aqt1LP * Note how this is related to [Function-orthogonality](Function-orthogonality.md) * [Fourier Transform](Fourier%20Transform.md) --- References * [Intro to the Laplace Transform & Three Examples](https://www.youtube.com/watch?v=KqokoYr_h1A) * [What does the Laplace Transform really tell us? A visual explanation](https://www.youtube.com/watch?v=n2y7n6jw5d0&t=522s) * [Fourier 3b1b](https://www.youtube.com/watch?v=spUNpyF58BY)