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### Course: Differential equations>Unit 3

Lesson 3: Laplace transform to solve a differential equation

# Laplace transform to solve an equation

Using the Laplace Transform to solve an equation we already knew how to solve. Created by Sal Khan.

## Want to join the conversation?

• Is there a known good source for learning about Fourier transforms, which Sal mentions in the beginning? I can't find it on Khan Academy.
• I made the video I always wanted to watch about Fourier series, I am going to put the link to the first part here

• Hey Sal, at you recommend to learn about Fourier series and Fourier Transforms. I tried to find videos about this but I didn't find any.
• Can you please do videos on Fourier series?
• Hello.

If it is ty "(t), we apply twice the formula of the derivative (to obtain the TL of y"), then the formula of multiplication by t. As you might have thought yourself.

Good work !
(1 vote)
• Sir can you please do video on the heavy-side function as well please
• *heaviside functions named after Oliver Heaviside. Pronunciation is the same though.
• what is fourier transform
• Sir,in the beginning of the problem you've applied the laplace transformation operator on both sides are there any precautions for that step, like you know, any necessary conditions or something
• There is an axiom known as the axiom of substitution which says the following: if `x` and `y` are objects such that `x = y`, then we have `ƒ(x) = ƒ(y)` for every function `ƒ`. Hence, when we apply the Laplace transform to the left-hand side, which is equal to the right-hand side, we still have equality when we also apply the Laplace transform to the right-hand side by the axiom of substitution. Is this what you had in mind?
• What if we don't know the values of y(0) and y'(0) and instead know other values of y and y', for example, y(1) or y'(1) or something alike. How would we solve the math ?
• Your videos have helped me to understand transforms much better than when my D.E. teacher was explaining them, and I've really come to appreciate them. So my question is, can I use Laplace transforms for all of these types of equations, or is it better to look at each problem as a case by case basis, and determine which method to use from there? Sorry if this sounds confusing...
• why is it that L(0) = integral 0 between 0 and infinity equal to zero
but the indefinite integral of 0 equal to a constant?
(1 vote)
• No, he's not saying the integral of 0 = C.
That constant comes from the other side of the equation. That integral = something + C and then he subtracts C from both sides.