ODE Exercise #5 - FIrst time, non-homo :v

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In today's episode we are going to deal with a non-homogeneous first order ordinary differential equation for the very first time! :) We are just going to apply what we have learned abstractly before to deal with this boi :3

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[weerman]

An 'e-schlange' a day keeps the doctor away

[suhailmall]

I used an integrating factor of exp(-4x) to get the same result

[holyshit]

This is linear first order, the special case of Bernoulli
Bernoull used variation of parameters to solve this equation

[bayhales]

Quicker but more risky way is to guess particular solution of form
y = e^5x*(Acos3x+Bsin3x). One of the cases where the guessing method comes in handy

[hOREP]

I tried using laplace transform to solve this, 3/10 would not recommend

Took ages and it went wrong *somewhere*.

EDIT: Aww man, I was so close too!

I ended up with y = 73/50 e^(5x) cos(3x) + 3/10 e^(5x) sin(3x) + (9/10 + f(0))e^(4x)

All that's different is that first bit, should be 1/10

[Soundillusionsxyz]

Non-homo? Wait... but I thought... what?

*siiigh* never mind, thought I got lucky

Anyway, FATHER, been having a bit of a rough time with differential equations, my teacher sucks as usual so a little video series on differential equation topics ranging from wronskians to the laplace transform would be welcome. <3

[riakm]

I opted for using Laplace Transform and it was relatively EZ because while there are 3 poles to evaluate the residue at, they are all first order so no messy differentiation to deal with.

[PackSciences]

Alternative solution for the integral I (at 5:00).
e^x cos(3x) = Re(e^(x(1+3i))) with i the imaginary unit.
int e^(x*(1+3*i)) dx = (1/(1+3*i)) * e^(x*(1+3*i)).
Now note that (1/(1+3*i)) = 1/10 - 3/10 i
Then, take the real part:
I = 1/10 * Re(e^x*(1+3*i)) - 3/10 * Im(e^x(1+3*i)) and you correctly get the same result as Flammable Maths found, with e^(a+b*i) = e^a * cos(b) + e^a * sin(b) * i.

[mokoufujiwara]

First step: integrating factor e^-4x, only took me a millisecond to think.
Second step: using complex number to do the tedious integration.

[rome]

Awsome Papa... Can you make a video proving the variation of variable stuff ?😁😁 it would be fun.

[Absilicon]

I just learned about solving second order non homogeneous differential equations and I have no fucking clue about how it works 😂. This is a linear one tho right?

[quahntasy]

"Homogenous" ain't no homo.

[crosisbh]

Hi I love your videos. Can you do more physics videos? They're my favorite from you.

[duncanw]

6:03 do I detect an engineer? 9=g y my boi

[rishabhdhiman]

5:15, its easier to use the complex form of cos(x)

[spacejunk]

I am not even done with basic Analysis. What am I doing here?

[omarifady]

Integrating factor is much easier ( e^-4x ) !

[MoJohnnys]

No homo, but I see John Forbes Nash Jr. in you. Just the "talking to people who are not there" part, though.

[hc_]

For the integral sonved with DI method, just try e^(a+bi) and get the general solution ;) It throws an interesting formula, you only need to know if it's the real part or the imaginary.

[StreuB]

I have no idea what is going on here, but thats ok because in another 2 semesters, I will. That is, if I pass calc2 and calc 3 the first time, which is highly unlikely. That you for your effort!