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## Differential equations

### Course: Differential equations>Unit 3

Lesson 3: Laplace transform to solve a differential equation

# Laplace/step function differential equation

Hairy differential equation involving a step function that we use the Laplace Transform to solve. Created by Sal Khan.

## Want to join the conversation?

• Am I wrong in thinking at the very end, in speaking of simplifying, that he could actually simplify the sin(t-2pi) and sin(2(t-2pi)) into sin(t) and sin(2t) because sin(t) and sin(t-2pi) will return the same value?
• You're right. Then one could collect terms for sin(t) and sin(2t), and pull out the common 1 - u(t-2pi). Furthermore, one may notice that the last factor is simply 1 for t less than 2 pi and zero afterwards, and thus we could write the result as:

``sin(t) / 3 - sin(2t) / 6 for t less than 2 pi and 0 otherwise``

This may even give you some insight into the equation -- t = 2 pi is the moment that the forcing stops (right-hand side becomes zero), and it so happens that the system is at an equilibrium at this time, so the response disappears too.
• In this video you have said that you have made a whole playlist on interpretation of differential equations at 0.45 but I am not able to find it anywhere...Cananybody please post the link..
• He actually said that he SHOULD do a playlist on the interpretation of differential equations, not that he has made such a playlist.
• can Laplace transformation give general solution of differential equation?
• You could, I think, arbitrarily set y'(0)=c1 and y(0)=c2 and proceed from there. The hairy algebra would be a lot hairier, but you'd get an answer eventually. And no, I'm not volunteering to work it through! :-)
• I have a question about relating Laplace to forcing functions. I know how to solve for an initial value problem. I know what my solution and forcing function is and I can graph both, but how are they related? Is the forcing function not allowing the solution to go past certain boundaries?
• I'm not entirely sure I understand what you're asking, but assuming you mean in a situation similar to that in this video.

The problem in this video is, if we look just at the homogenous part, a mass on a spring problem. The angular frequency of this undriven system is 2.

The forcing function (here sin(t)-u2pi(t)sin(t)) can be seen as some additional driving force which in this case is, up to 2pi seconds (two oscillations of the undriven, homogenous system), giving the system the acceleration it would need to oscillate with an angular frequency of 1.

The driving force, is trying to force the oscillation to go at half its natural frequency.

Furthermore, because the mass starts at the origin and at rest, the driving force serves to actually get the thing moving because without it it would quite happily sit around doing nothing all day.

Because the two oscillations (the natural sin(2t) and the driven sin(t)) are in phase, and 2pi is a integer multiple of a half-period for each of them, when the step function toggles high and the driving force stops, there isn't any force to keep the mass moving and so it too will stop.
• Isn't sin(t-2pi) the same as sint? Sin is periodic over 2pi.
• Indeed. Both give the same value for an arbitrary value of t.
(1 vote)
• At , shouldn't the f(t) in blue be f(s) instead?
• What textbook does Sal use for differential equation problems?
• is u subscript 2pi the same as u(t+2pi)?