Differential Equations with Forcing: Method of Variation of Parameters

This video solves externally forced linear differential equations with the method of variation of parameters. This approach is extremely powerful. The idea is to solve the unforced, or "homogeneous" system, and then to replace the unknown coefficients c_k with unknown functions of time c_k(t), and then solve for these functions based on the external forcing.

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This video was produced at the University of Washington

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[akanguven]

May the Force be with you, GOD bless you Professor Brunton. Thank You !

[hoseinzahedifar]

Great lecture...Such a useful scheme to solve 2nd Differential Equations...Thank you very much, Steve...❤🧡💚

[kostasdiamantis]

Great video. Just an observation, you could easily solve u2 by rewriting the Condition 2 as -u'1*e^(-t) - 2*u'2*e^(-2t) = -[u'1*e^(-t) + u'2*e^(-2t)] - u'2*e^(-2t) . The [ ] equals 0 as we assumed so. Great series!

[drskelebone]

This was one of my favorite parts of my differential equations class.

[mathjitsuteacher]

Hi Steve, only a small correction. Many times in this video you say Chain Rule when what you are using using is the Product Rule.

[juanmanuelmillansanchez]

Hi Steve, thanks for all the content, just dont forget to update the differential equation playist.

[lgl_noname]

I found what I had missed.
thank You.

[Eta_Carinae__]

I've always wondered about VoP: if you don't assume C1, is there a class of solutions seperate from the one you got here, or is C1 a necessary feature of a solution? Thanks for the video. VoP is a mouthful of work!

[qchentj]

Thanks for the good course. Just not sure why suddenly C1 assumption came out of nowhere.

[APaleDot]

I'm confused. When he takes the derivative of x he uses the product rule, but when he take the derivative of x' he doesn't?

EDIT: oh wait, I think I get it now. He assumes the u' terms add to zero, so you don't get any u'' terms in the equation for x''

[marc-andredesrosiers]

I recognize the = 0 step works.
Not sure I get when it's OK or not to make the assumption.
In particular, did we prive we found all solutions of the equation[?]

[JoseMarcos-ljwc]

Dear professor, thank you very much for your lectures. I learned a lot. And I am really interested on the topic, however some of the uploaded videos are unavailable to me (12/45, to be precise). Is it what you intended for them to be or is it a problem with youtube? Thank you again.