You can rewrite it as follows: y' - xy'' = - (y')^3 Then divide both sides by (y')^2 with a remark that y' = 0, which means y = const, is a solution. (y' - xy'')/(y')^2 = - y' (x/y')' = (-y)' x/y' = - y - c/2, where c is a constant x*dx + (y + c/2)*dy = 0 x^2 + y^2 +cy + a = 0, where "c" and "a" are real constants.
at 2:18 mark, all you had to do was to compare the second degree polynomial coefficients to 1 . it directly gives you { A+B=0 , C=0 , A=1 } which implies B = -1
Barnes (of the Barnes G-function) was an interesting character. IIRC, he was a bishop and, at the time, this automatically granted a seat in the then dominant House of Lords. He became embroiled in a verbal altercation with a fellow peer to which Barnes responded with potentially defamatory comments. However, parliamentary privilege was then as it is now and provides protection to members against all legal action along the lines of libel, slander and the like. Anyway, Barnes’ nemesis reportedly challenged our man to repeat his comments outside the walls of Westminster, which he duly did, to the riled up press corps no less! The result was a predictable lawsuit, the result of which was likely quite damaging given that his name apparently lives only in regards to his discovery of the eponymous G-function. Anecdote courtesy of my 1st year Mechanics lecturer who, many years later, visited London to give a guest lecture on his more recent forays into Number Theory…specifically, applications of Random Matrix Theory. And that’s probably sufficient information for one comment!
Hey, just wanted to let u know that there is an easier way to solve the LHS of the equation at 1:28 I don't remember where, but I saw a trick to solve integrals of that form. Just take out a common factor of u^3 from the denominator and move it to the numerator. Now the numerator is the derivative of the denominator. Hope that helps
Bruh the only reason I put it there is cuz of you 😂 You once pointed out that I should be careful about sqrt(x²) and you're right and its stuck with me ever since and will stick forever 😂😂😂
That being said you actually have a huge influence on my work. I didnt intend to teach anyone through my videos and then you started nitpicking over little details that should be made clear for those trying to learn the kind of math I use. So I started to explain stuff more and more even when the details seemed trivial to me. So thanks bruh
@@maths_505 I'm thinking of making a list of infinite series that can be evaluated using different Fourier expansions (for example (-1)^n/(2n+1)^3 whose value can be found from Fourier sine series of pi*(pi-x) over 0 to pi). If you had some extra free time, I would like us to collaborate.
The second channel will have formal lectures on complex analysis and differential equations. I'm currently working on lecture notes and I'll add more courses with time.
For the part 1/u(u^2 + 1), I guess you're trying to make it formal. Normally I'll just do: 1/u(u^2 + 1) = (u^2 + 1 - u^2) / u(u^2 + 1) = 1/u - u/(u^2 + 1).
Question to Maths 505, can i send you an indefinite integral on instagram (i follow you thats not a problem). I tried the same DE with (y')⁴ instead and i have to solve it to get y but i cant..
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You can rewrite it as follows:
y' - xy'' = - (y')^3
Then divide both sides by (y')^2 with a remark that y' = 0, which means y = const, is a solution.
(y' - xy'')/(y')^2 = - y'
(x/y')' = (-y)'
x/y' = - y - c/2, where c is a constant
x*dx + (y + c/2)*dy = 0
x^2 + y^2 +cy + a = 0,
where "c" and "a" are real constants.
at 2:18 mark, all you had to do was to compare the second degree polynomial coefficients to 1 . it directly gives you { A+B=0 , C=0 , A=1 } which implies B = -1
Pretty cool and simple equation! Bring more differential equations to the channel❤
SUIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
have you ever done a video on the Barnes G function or the hyperfactorial function? they are quite silly and fun and get you some cool results
That's a wonderful suggestion
@@maths_505 you can actually integrate ln(G(x)) or any case of x^n*ln(G(x)) and get some goated results
Barnes (of the Barnes G-function) was an interesting character. IIRC, he was a bishop and, at the time, this automatically granted a seat in the then dominant House of Lords. He became embroiled in a verbal altercation with a fellow peer to which Barnes responded with potentially defamatory comments. However, parliamentary privilege was then as it is now and provides protection to members against all legal action along the lines of libel, slander and the like. Anyway, Barnes’ nemesis reportedly challenged our man to repeat his comments outside the walls of Westminster, which he duly did, to the riled up press corps no less! The result was a predictable lawsuit, the result of which was likely quite damaging given that his name apparently lives only in regards to his discovery of the eponymous G-function. Anecdote courtesy of my 1st year Mechanics lecturer who, many years later, visited London to give a guest lecture on his more recent forays into Number Theory…specifically, applications of Random Matrix Theory. And that’s probably sufficient information for one comment!
awesome as always, congrats on 30k bro🎉
Very interesting differential equation, thanks a lot👍
Hey, just wanted to let u know that there is an easier way to solve the LHS of the equation at 1:28
I don't remember where, but I saw a trick to solve integrals of that form. Just take out a common factor of u^3 from the denominator and move it to the numerator. Now the numerator is the derivative of the denominator. Hope that helps
Excellent
Comparing coefficients in your partial fraction calculation can produce some really simple equations for A&B, say.
It is pretty to solve it in simplest way. Thanks.
In the final result, you can bring the 1/c term inside the root, to get [ the root of (B-x^2)+c2], wher B is another constant.
You don't need the +- in the final answer (or the intermediate steps) because C is already a free parameter that can be positive or negative.
Bruh the only reason I put it there is cuz of you 😂
You once pointed out that I should be careful about sqrt(x²) and you're right and its stuck with me ever since and will stick forever 😂😂😂
That being said you actually have a huge influence on my work. I didnt intend to teach anyone through my videos and then you started nitpicking over little details that should be made clear for those trying to learn the kind of math I use. So I started to explain stuff more and more even when the details seemed trivial to me. So thanks bruh
Using Bernoulli's would make it a bit faster than pfd
Yes indeed
@@maths_505 I'm thinking of making a list of infinite series that can be evaluated using different Fourier expansions (for example (-1)^n/(2n+1)^3 whose value can be found from Fourier sine series of pi*(pi-x) over 0 to pi). If you had some extra free time, I would like us to collaborate.
Sir can u please tell me which course are you providing on here after I want to be a scientist Im able to tell everyone that I'll joined that channel
Well it's not really a course I'm just doing applied math for fun
That was smooth
First DE i do myself and then check the result after watching a mini series of u solving DEs. Nice one
SUIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
The second channel will have formal lectures on complex analysis and differential equations. I'm currently working on lecture notes and I'll add more courses with time.
@@maths_505Wait, you will have a second channel?? Fucking great whats the name
@@ΙΗΣΟΥΣΧριστος-θ2γyoutube.com/@TheHouseOfMathness505?si=iT14xDPNOz6PhA44
@@ΙΗΣΟΥΣΧριστος-θ2γif the link doesn't show up/gets deleted by TH-cam with it's stupid rules for links, here's the tag @TheHouseOfMathness505
For the part 1/u(u^2 + 1), I guess you're trying to make it formal. Normally I'll just do: 1/u(u^2 + 1) = (u^2 + 1 - u^2) / u(u^2 + 1) = 1/u - u/(u^2 + 1).
What application do you use to write all of these? To be more general, what is your setup for doing your videos?
My phone and stylus
I wish you had pushed one step farther for the implicit formula, because it's especially pretty.
Look who wants an implicit solution 😂
y = C is also answer
Arctan u?
This implies x100
Noice
Question to Maths 505, can i send you an indefinite integral on instagram (i follow you thats not a problem). I tried the same DE with (y')⁴ instead and i have to solve it to get y but i cant..
Sure