An interesting case study in differential equations

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Пікірлер: 42

@renesperb 10 ай бұрын

The differential equation y' ' =c* y^ p , together with boundary conditions y '[0]= 0 , y[xo] = 1 plays a role e.g. in chemical engineering . Here p can be any real number. In particular , for 0 < p

@pandavroomvroom 10 ай бұрын

math 505 with 50.5k subs

@o0QuAdSh0t0o 10 ай бұрын

🤤🤤

@Dravignor 10 ай бұрын

Lezgooooooooo

@JeeAspirant2024-j7f 10 ай бұрын

50.6 now 😢

@humhum3987 10 ай бұрын

Road to 505k subs

@edmundwoolliams1240 10 ай бұрын

If this were a university lecture, the N=0 and N=1 cases would be explained painstakingly thoroughly for 45 minutes and explained how useful the applications are. Then the N>1 case would be haphazardly scribbled a little bit then left as an exercise

@maths_505 10 ай бұрын

😂😂😂

@maths_505 10 ай бұрын

Well I guess that's why we love KZbin math😂

@edmundwoolliams1240 10 ай бұрын

@@maths_505 Exactly ;P

@tomaszkochaniec9421 10 ай бұрын

More difficult is y^n=n-th derivative with y eg. y^n=y^(n)...

@coreyyanofsky 10 ай бұрын

interesting to contemplate substituting N = 0 or 1 into the general solution N = 0 recovers that specific case directly N = 1 requires taking a limit

@maths_505 10 ай бұрын

Exactly the reason I wanted to start with them separately as sort of a buildup towards the general solution.

@MrWael1970 4 ай бұрын

Nice idea. Thanks

@Calcprof 10 ай бұрын

n=2 can be expressed in terms of the Wierstrass Pe function. The y'' == (y')^n version is easier, at least to get some solutions.

@maths_505 10 ай бұрын

Yes indeed

@merwan.houiralami 9 ай бұрын

it’s so unsatisfying to be left with such solutions and not clear regular ones, and this for so many equations !!!

@michaelbaum6796 10 ай бұрын

Cool solution👍

@edmundwoolliams1240 10 ай бұрын

Omg I've been waiting so long for someone to do this!

@euler1 10 ай бұрын

I don't understand what you did @ 2:13, how could you change the second derivative of y to (y'/2)^2 ?

@aravindakannank.s. 10 ай бұрын

im lucky to explain it to the great Euler the thing is he reverse engineered differential by integrating it to achieve it 😊with respective to the x on both sides

@juliavixen176 10 ай бұрын

It's the integral (the antiderivative) of y'y'' which then has the derivative operation written separately in front of it. So when you do (or redo) the derivative of the antiderivative you get back your original y'y'' In Leibnitz notation, (dy/dx)(d²y/dx²) when integrated for (dx) gives you ( (1/2)(dy/dx) )² and if you then later apply a (d/dx) to that, you get (dy/dx)(d²y/dx²) back... I think. (Disclaimer: I might be wrong. I'm not confident in my ability to solve differential equations right now.)

@aravindakannank.s. 10 ай бұрын

@@juliavixen176 u r just right keep ur confidence up😊

@bowlteajuicesandlemon 10 ай бұрын

Using u-subs, if you have y'*y", then you can take y' as u, using the concept integral(f(g(x)g'(x)) = integral(f(u)) w.r.t. u, F(u). So you end up with integral(u) = u²/2 = (y)²/2

@marcellobernardini8412 9 ай бұрын

It would be intresting a video where you analize also N

@hortlegende 10 ай бұрын

Why can you simplify (+-x)^2/1-n as (x^2)^1/1-n? For n=3 you have +-x^-1 and absolute value of x ^-1, maybe (probably) forgetting something (7:00)

@aravindakannank.s. 10 ай бұрын

either it's positive real number or negative real number by squaring it we will positive squared number of real number same goes with algebra😊

@stophel_en_blick7612 10 ай бұрын

can you split the fraction in the exponent like in the last part ? because for instance we have cases like (-1)^{2/6} = (-1)^{1/3} = -1 eq 1 = 1^{1/3} = ((-1)^2)^{1/3}.