non-diagonalizable systems -- differential equations 22

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[frankjohnson]

Would love to see the bonus content you mentioned about the general method using matrix exponentials

[HeriksYT]

Thank you for videos. I really enjoy them. :)

However, I want to point out, that at 17:16 you wrote 5*c1+ 5*c2*t = b1 * (1+t), where b1 = 5*c1+ 5*c2
I don't think this is correct, since this Product produces extra terms in 5*c1*t and 5*c2

[edl]

This is AMAZING! Almost all answers just smash the result on your face w/ telling where it comes from

[AriosJentu]

On our lessons we were taught (and also I'm teaching my students) a little bit different techniques. And your answer C1(1+t) v1 e^kt + C2 v2 e^kt confuses me.

Since the roots of characteristic poly are the same, and also they are linearly independent, we should calculate all available attached vectors for this eigenvalue in the next form: (A-kI)*v_j = v_{j-1} (based on the previous vector, but it's the same as: (A-kI)^j v_j = 0.
If we use increasing order techniques to increase from n linear equations to 1 n-th order linear equation, integrate, and substitute as vector forma, we get something like this:
y = C1 v1 e^kt + C2 ( v1 * t + v2) e^kt + C3 (v1 * t^2/2 + v2 * t + v3) e^kt + ...
We called it "Solution family generation" - first generation is a single person "v1", but on the second generation it has a first corresponding attached vector "v2", which means that at this time vector "v1" is a "faTher", and "v2" is a son - we multiply "v1" by "t", and add it's son: v1 * t + v2, and on the third generation "v1" is a father of father, "v2" is a father, and "v3" is a son. Grandfather vectors we multiply by t^2/2, etc. Every next step of linear dependendent repeated eigenvalues corresponding integrating the previous one by "t", and as integrating constant vectors we got the next linear dependent eigenvectors. Integrate "v1" we get "v1 * t + v2", integrate this again, we get "v1 t^2/2 + v2 t + v3" and etc.

There is a mistake in this video, which confuses me: when we replace "C1 + C2" with "B1", construction "C1 v1 + C2 t v1" we haven't any possibility to write it as "B1 (1+ t)", and it's explained in this comment on text above.

Thank you)

[trtlphnx]

Thanx For Sharing Your Hard Earned Knowledge; You Are The Best~

[holyshit]

y'=Ay+B
y_{p} = sum(c_{k}(t)φ_{k}(t), k=1..n)
c'(t)=Φ^{-1}B
where Φ is fundamental matrix and φ_{k} is kth column of fundamental matrix
Wronskian can ce calculated as follows W = C_{0}exp(Int(tr(A), τ = t_{0}..t))

[edl]

Oh just realized a mistake in the middle lol

[synaestheziac]

Here’s what I got for the warm-up:

*y* = c₁(1 + t)e²ᵗ(1, -1) + c₂e²ᵗ(0, 1)

Anyone want to check that?