Multiply the differential equation by e^(-t). After that the left hand side can be written as (e^(-t)y)''. The right hand side simplifies to e^(-5)δ(t-5). Then you only have to take antiderivative a couple of times, knowing that δ = H' and H = (tH)'. Before using the initial conditions, you get y(t) = (t-5) e^(t-5) H(t-5) + At e^t + B e^t. Initial conditions make A = B = 0.
Hey Dr have do you know of Bell Polynomials for repeated differentiation of the compound function f(g(x)) called the Faa di Bruno formulas generalísimo the chan rule to higher derivatives . Maybe do a video on them , they are beautiful
Multiply the differential equation by e^(-t). After that the left hand side can be written as (e^(-t)y)''. The right hand side simplifies to e^(-5)δ(t-5).
Then you only have to take antiderivative a couple of times, knowing that δ = H' and H = (tH)'.
Before using the initial conditions, you get y(t) = (t-5) e^(t-5) H(t-5) + At e^t + B e^t. Initial conditions make A = B = 0.
Oh, this one brings back memories of the theory of systems and signals at uni.
Nice video Peyam. Have a great 2025!
Amazing derivation Dr.
This example nicely shows the strength of the L-transform for these types of problems .
7:57 Number 5 is alive!
Of course variation of parameters can be used to solve this! Let me know if you couldn't figure it out.
are you from Tabriz ?? pls Answer :)
Yes he is
@BrendanLawlorSTUD how did you find out ?
@nimaalz4513 he told me a while ago
@@BrendanLawlorSTUD finally :)
He told me @@nimaalz4513
Hey Dr have do you know of Bell Polynomials for repeated differentiation of the compound function f(g(x)) called the Faa di Bruno formulas generalísimo the chan rule to higher derivatives . Maybe do a video on them , they are beautiful
What’s the U function called?
Heaviside step function
@ oh thanks I’ve seen that with an H not a U