Differential Equations thread How in the fuck does one apply

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This is what I wrote down but I must have fucked up somewhere because I got the denominator right but the numerator is fucked up

Wouldn't you set t-6 as a standard t and then do it out from there?
Like u(t-6) would be u(t) and then sin(4t-24) would be sin(4t) and then plug back in later or something?

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>>8536375
Yeah I noticed that too, but I just don't understand what to do with the t and e^2t, I was thinking that I use the two equations t^n f(t) and e^at f(t) and using the u(t-6)sin(4t-24) as f(t)

>>8536386
Use t^n f(t) with u(t-6) sin(4(t-6)) as your f(t). Then use e^at f(t) but now f(t) is the whole thing t^n u(t-6) sin(4(t-6)).

>>8536402
Yeah just realized that's what I did in the first place except I used e^at first and I just fucked up the derivative and didn't simplify it

Brainlet thread.

>>8536341
u(t-6)*t*exp(2*t)*sin(4*t-24)
= u(t-6)*((t-6)+6)*exp(2*(t-6)+12)*sin(4*(t-6))
= u(t')*(t'+6)*exp(2*t'+12)*sin(4*t') where t'=t-6

u(t-a)*f(t-a) -> e^-as.F(s)
f(a*t) -> (1/a)*F(s/a)

After that, I'd convert sin(x) to (e^-ix - e^ix)/2..

Anyhow, it all ends up as

8*(3*s^2-11*s+58)*e^(12-6*s)/(s^2-4*s+20)^2

>>8536429
When I did it I got
8(e^12)(e^-6s)(s^3-6s^2+12s-37)/(s^4-8s^3+40s^2-128s+400)
I haven't done this in years so don't trust me.

>>8536433
That's the correct answer or at least close I believe, but you expanded the bottom

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I don't get this one, tried every method

>>8536469
Change s^2 + 4s + 20 into (s + 2)^2 + 16, then use partial fraction decomposition, and then modify the numerator of the (s+2)^2 + 16 term to have the s+2 shift.

>>8536341
Not OP but I also have a question?

Why is all differential equations stuff focused on 1st and 2nd order ODEs?
Why not 3rd order?
I do grad researched and they come up pretty often.

>>8537239
Because Newton's second law is 2nd order.

>>8537239
anything higher order than 2 is a pain in the ass to get an analytical solution out of. 9/10 3rd+ order DE's are solved with Rung Kutta or some other numerical method.

>>8537239
>Why is all differential equations stuff focused on 1st and 2nd order ODEs?
it doesnt? e.g. for constant coefficients a simple solution theory for ODEs (for initial value problems) of arbitrary order exists.

>>8537239
Usually 3rd order or higher derivatives don't have much practical use (usually but not always), so there's not much reason to go into it but methods like constant coefficients can be used on such linear equations

>>8537265
Yes, I know.
However, third orders still appears a lot.
Mainly in the fields of thermodynamics, gases, diffusion, etc.