Second-Order Ordinary Differential Equations: Solving the Harmonic Oscillator Four Ways

Here we introduce the second-order ordinary differential equation (ODEs) for a mass on a spring. In Newton's Second Law, F=ma, the acceleration a is the second derivative of the position x(t), giving a second order differential equation. We solve this equation four ways: 1) by guessing the solution, 2) using Taylor Series, 3) by guessing a different form of the solution, and 4) by writing as a matrix system of equations.


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This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Introduction
1:00 Deriving the Spring-Mass Equations from F=ma
6:59 Method 1: Guess the Solution!
11:57 Method 2: Taylor Series Solution
24:18 Method 3: Guess Again!
33:20 Method 4: Write as a Matrix System of Equations

[schenzur]

I'm loving this whole series, thanks a lot professor for putting forward a whole course!

[bitterthread]

Keep uploading the content Professor. I am taking Engineering courses and this has been very useful for even providing motivation how all things are fundamentally related .

[hoseinzahedifar]

A perfect topic by an excellent teacher... thank you.

[Pedritox]

Great video! Would be great videos of Lagrangian's dynamics

[ErikScott]

This is how I wish differential equations was taught when I was in undergrad, and methods 1 and 3 are the approach I'm taking for a video I want to make on rotorcraft flapping.

Around time 31:42, you start trying to solve c1 and c2 in terms of the initial values x0 and v0. My response to this is, why bother? You already established that c1 and or c2 must be complex valued in order to make the final equation real. They are arbitrary constants, and i is a constant as well. So, when I demonstrate this, I just say "let a=c1+c2, let b=(c1-c2)i" and then establish a=x0 and b=v0. No need for complex analysis, and the result matches the intuitive guess (assuming you were able to guess the sine component) and the Taylor series result, while still being completely general.

[FA-tqip]

Around 17:28 I think you left the minus sign out on the board which I have put in single quotes (although you do say it): => C3 = '-'1/3! * V0 - (from an example I calculated it also seems like there should not be a minus here: => C5 = '-'1/5! * V0)

Seems like magic. Thank you very much!

[Buyson]

What about Laplace Transform? I think we could use it as well.

[salahbeed]

28:00 ✨ok that was just the fire alarm✨

[riteshparmar]

Thanque very much for this beautiful lecture.

[halilibrahimEgilmez]

in time 30:00 if λ is complex like a±ib then the answer should be x=e^at (x_0 cos(⁡bt) + v_0 sin⁡(bt) ) and here a=0 and b=1 then x=x_0 cos⁡t+v_0 sin⁡(t)

[curtpiazza]

I learned so much in less than 40 min.! Thanx! ❤ 😂

[VTdarkangel]

There is also the characteristic equation. That being said, I suspect that he is going to hit in that with the eigenvalue determinant or if he covers Laplace Transforms.

[timgorringe]

great lecture! is x_o cos(t) - v_o sin(t) soln dimensionally incorrect? (taylor series soln)

[danieljulian]

Linear operator! Superposition! Great! 27:50 So, I guess we're assuming some linear algebra already. Not the first time in this series that I've seen messages from the future.

It's not that free stuff is useless, but anyone watching these is sinking a half-hour at a time into following the presentation. It seems more that these videos are intended as refresher, or as quick review of formal lecture material given in a physical classroom by Steve himself. Anyone else is going to have to forego any pretense of rigor or fill in a lot of blanks with a lot of legwork. This series is not a substitute for a complete course.

[arthurcoward]

So how do you get the notes you're writing to appear correctly? That is, writing right to left when I'm pretty sure you would have to be writing backwards to achieve that since you're facing the viewer. Enjoying the talks, btw!

[aris.mavridis]

I'm having a bit of trouble using the second method for any k/m instead of just 1.
Would the Xo and Vo terms be multiplied by k/m as well?
I tried to find such solutions online but had no luck.

[seannibecker]

Are you actually writing left to right from your perspective?

[muthukamalan.m]

how do we visualize second order DE equations like we understand by seeing slope field in 1st order DE ?

[benarcher]

Strange. In Method II the answer is x(t) = cos(t)x_zero + sin(t)v_zero. But here we are adding a length to a velocity!? The answer x(t) is a length. Perhaps the second term should have been multiplied by 't'? Am I missing something here?