8: Eigenvalue Method for Systems - Dissecting Differential Equations

When we start looking at how multiple quantities change, we get systems of differential equations. What do we use for systems of equations? Linear algebra, of course!

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[blackpenredpen]

Believe it or not, I have forgotten this method! 😆

[ellen]

Fully understand it now. Math shouldn't be about memorizing formulas. It should be about connecting the dots, observing patterns, and summarizing patterns into shortcuts called formulas, and enjoying the thinking that goes into that process.

[santiagoarce]

You can really tell that there's a lot of thought put into these videos about what might be the best way of explaining something. I actually find your way of explaining stuff really helpful.

[RedSarGaming]

What a great video! Short and explains everything clearly. I cannot thank you enough my guy!

[BuddyNovinski]

Having taken both differential equations and linear algebra a long time ago, I can see the link between the Wronskian and vectors. How I wish this young man had been my professor (in his previous life). We have the advantage of the internet, so it's much easier to learn this stuff.

[elischrag]

this was super helpful! Made the connection between eigenvalue/vector problems and solving differential equations much more clear.

[IntegralMoon]

I found myself confused by MIT OCWs explanation, but this cleared it right up. Lovely work

[holyshit]

In russian textbook i found following method
x' = 2x + 3y
y' = 4x + y

x' = 2x + 3y
ky' = 4kx + ky


x' + ky' = (2+4k)x + (3+k)y
x' + ky' = (2+4k)(x+(3+k)/(2+4k)y)

(3+k)/(2+4k) = k

3 + k = k(2 + 4k)
3 + k = 2k + 4k^2
4k^2 + k - 3 = 0
(4k - 3)(k + 1) = 0

x' - y' = -2x +2y
d(x - y)/dt = -2(x-y)
d(x - y)/(x-y) = -2
ln(x - y) = -2t+ln(C_{1})

x - y = C_{1}exp(-2t)

x' + 3/4y' = 5x + 15/4y
d(x + 3/4y)/dt = 5(x+3/4y)
d(x + 3/4y)/(x+3/4y) = 5dt
ln(x + 3/4y) = 5t + ln(C_{2})
x + 3/4y = C_{2}exp(5t)


x - y = C_{1}exp(-2t)
x + 3/4y = C_{2}exp(5t)

3/4x - 3/4y = 3/4C_{1}exp(-2t)
x + 3/4y = C_{2}exp(5t)
7/4x = 3/4C_{1}exp(-2t) + C_{2}exp(5t)
x = 3/7C_{1}exp(-2t) + 4/7C_{2}exp(5t)

x - y = C_{1}exp(-2t)
-(x + 3/4y = C_{2}exp(5t))
-7/4y = C_{1}exp(-2t) - C_{2}exp(5t)
y = -4/7C_{1}exp(-2t) + 4/7C_{2}exp(5t)


x = 3/7C_{1}exp(-2t) + 4/7C_{2}exp(5t)
y = -4/7C_{1}exp(-2t) + 4/7C_{2}exp(5t)


To generalize it for more equations we need k1, k2..., k_{n-1}
but problems may appear for repeated eigenvalues

I dont speak russian (When i went to school it hadn't been taught. My mother didnt want to teach me)
but this approach probably has something to do with eigenvalues and eigenvectors

[aijazdar]

But 'r' is actually 'A' SO BOTH ARE SAME, HOW U ARRIVED EIGEN VALUE EQUATION

[rodioniskhakov]

3:48 I don't understand why do you say that x' = e^(rt) but not x' = e^(At). Why r = A? Or rather, why e*I = A?

[troyhernandez]

Man, I forgot my differential equations classes. Very cool method.

[duydangdroid]

if my professor ever did this, i'd say "whoa... an easy day"

[drseagull]

what a god, you summarised a 30 min lecture in under 10 mins

[dulosalfred]

This is so informative. Thank you for a comprehensive explanation.

[shadowbane]

it is most definitely E I G E N V A LU E T I M E

[lukaskrause]

Does the superposition principle only work for homogeneous differential equations?

[FelipeHenrique-yqbu]

I got the eigenvector associated to the eigenvalue -2 equal to (1, -4/3), is this okay?

[izu]

It will take me a year to connect the dots😂🔫

[tonk]

Hi...I want u to add a video for solving system of diiferntial eqns hving complex roots by using matrix exponential form....

[chigbuchiamaka]

This is the only video I have watched explaining the concept behind solving system of equations 😩. Thank you Sir