Thanks! Interesting idea - would be interesting to see how 4D space could be visualised in this way.

Sadly, its not my choice to have ads -- I would prefer not to.

Thanks!

Thanks - I will be doing another video on vector spaces soon.

Thank you for the generous complement!

Thanks --- I agree. Let there be time stamps.

I missed this comment. In the theory of 'ordinary differential equations,' which is what we are doing here, we always assume that y is a function of a single variable x. So when we are given a DE like x dx + y dy = 0, we assume that y is a function that satisfies the equation above. Of course, we then want to solve the equation (find all possible y). Solutions are given in two ways: either by writing y as a function of x, or implicitly by writing y as a solution to a certain two-variable equation.

Thanks for this comment --- it is appreciated! It is indeed an error (which I should have picked up). Glad to hear the videos help.

Yes - any linear combination of solutions to a homogeneous differential equation (with 0 on the right) is itself a solution. Solving such a differential equation means finding all possible solutions. For this reason, finding one solution does not solve the differential equation. However, if we are given initial values, then we can single out one solution - the unique solution. For example, solving the equation x^2 - 1 = 0 means to find all possible x that satisfy the equation (x = 1, -1). For this reason, x=1 is not the solution; it is 'a' solution.

ahhh thank you I understand, now it makes sense why linear independance matters!

@@peradies7044 I am glad!

Which part?

he's asking about 28:50@@mhoefnagel92

Thanks, I am glad they help

The limit exists or not, depending on the value of s, hence the cases.

I’m glad to hear it :)

Glad to hear it!

I do aim to do that at some point

@@mhoefnagel92 I have NEVER seen a better intuitive explanation of the Laplace transform. Congrats.

@@DavidFMayerPhD many thanks!

@@mhoefnagel92 By the way, my PhD IS in Mathematics. When I went to school, there were NO SUCH VIDEOS. I would have learned more if my books had been supplemented by videos such as yours.

@@DavidFMayerPhD thanks for these generous remarks, I’m glad the receive them.

This is not an easy Laplace transform. There are a few standard tricks that I have seen --- using gamma function, as you say, but also using Laplace transform of 1/sqrt(t). There are a few videos on KZbin explain these and other approaches.

Yeah, pretty much right, things do turn out suprisingly well when you replace factorials with gamma functions

We apply the Laplace transform to an IVP, it becomes an algebraic equation, which can then be manipulated (often using partial fraction decompositions) into a form which we can apply the inverse Laplace transform (giving a solution to the IVP).

Consider this function: y= e^(Ax). To take the derivative, take the derivative of the outer function, which is e^(Ax) since the function is its own derivative, and multiply by the derivative of the inner function: Ax, or simply A and multiply: y'= Ae^(Ax). To find the antiderivative, integrate the outer function, again, e^(Ax) and divide by the derivative of the inner function so: Se^(Ax)dx= (1/A)e^(Ax). This means that taking the derivative is simply multiplication, and integration is simply division. It would be nice to be able to do that for general functions: y= f(x), or something like: y= sin(x). It's not generally the case that the derivatives and integrals are related by constants. Therefore, make your general function a function of e^(Ax) by multiplying with e^(Ax): e^(Ax)f(x). For this, make the substitutions: e^(-sx)f(x) where s is a new variable, and may be real of complex. Next, get rid of the 'x' term by integrating from zero to INF. If the integral converges, you have a new function that's a function of s only. This is your LaPlace Transform. If you can do integration by parts, you can integrate: S[f'(x)e^(-sx)dx and see that F'(s)= sF(s) - F(0) (F(0) takes care of initial conditions).

​@@lordkelvin100thompson8 Thank you.

Sure - I use a Python library for mathematical animations, which is called "manim", short for "Mathematical Animation engine". The mathematical formulas are all in LaTex (rendered with a short manim script).

235th

Thank you!

​@@mhoefnagel92 yes true you should