yeah. truth is x(t) = Ae^((+ or - )t) + 2npi - pi/2 when you take this into consideration. plugging it back in, you do get the equation to work still (even for the + case)
My main takeaway from all the experience I’ve ever had with differential equations is that if you simply guess that the solution is some variation of the exponential equation, you’ll probably be correct.
It’s not as hard, as like e^(dx). Just like inverse operation of exponential or trig derivative. We can use power series, as usual, to create close form for that
Neat differential equation! But how about “x = cos x”? The result is an irrational (and even transcendent!) number that might be worth a mention on your channel. :D
@@PapaFlammy69 And I just realized, you ay have probably done the derivative of x^x before, but can you take the NTH derivative of it in general? No recursion allowed!
Since the solution to cos(x)=y is x=(+ or -)arccos(y)+2*pi*k, I'm thinking you need a 'plus or minus' in front of the -x-pi/2.
yeah. truth is x(t) = Ae^((+ or - )t) + 2npi - pi/2 when you take this into consideration.
plugging it back in, you do get the equation to work still (even for the + case)
nice video... love the advent calendar series
My main takeaway from all the experience I’ve ever had with differential equations is that if you simply guess that the solution is some variation of the exponential equation, you’ll probably be correct.
How about integral sin(dx)
I would LOVE to see a step-by-step solution to this.
It’s not as hard, as like e^(dx). Just like inverse operation of exponential or trig derivative. We can use power series, as usual, to create close form for that
Nearly got through the Advent Calender! Keep up the work!
That equation could also be written cos(x') = (cos(x))' which feels like even more of a "freshman's dream".
yeah!
another weird cursed problem was solved🚬🗿
5:14 how come you can cancel the dt's?
It’s the opposite of the chain rule:
df/dt = df/dx•dx/dt
You can look up “Integration by substitution” on Wikipedia. It contains a complete proof.
it's called magic
however, -sin(x)=cos(-pi/2-x) or cos(pi/2+x) as cosine function is even, thus the o.d.e. to be solved is x'=2kpi \pm pi/2 \pm x (:
Great video great explanation
Proven the shit!
Has Papa Flammy finally integrated e^(x^2) yet? Or is he being an engineer about it?
Neat differential equation! But how about “x = cos x”? The result is an irrational (and even transcendent!) number that might be worth a mention on your channel. :D
already done :)
@@PapaFlammy69 Okay, then you were ahead of me. But maybe you can take that, make it 10 times harder and then show it off. :D
@@PapaFlammy69 And I just realized, you ay have probably done the derivative of x^x before, but can you take the NTH derivative of it in general? No recursion allowed!
@@trwn87 Just use the Faà di Bruno's formula for exp(x*lnx) and then the Leibniz rule for x*lnx.
You didn't "prove the shit" out of it, because you didn't differentiate e^-t the way that a 'real man' would.
Versteh ich nicht
very early