Hello. First of all great content on your channel, love it and keep it up! :) Secondly I am in the first year of my master degree, mechanical engineering. For some reason in all these years of maths classes we have never actually talked about Secant and Cosecant. I know about cos, sin, tan and cot. So my idea for next video: do a quick rundown for sec and cosec, basic definiton, usefulness etc? I guess I could google all that, but I just enjoy watching you explaining stuff since it's far more interesting. Good luck and thanks :)
Yes, this is a great video. Thank you! You should post more differential equation videos, too. You really do help with problem solving skills. Excellent.
If you know the limited development, this is quite easy: as you know tan (X) = sin(X)/cos(X) so we are looking for the limit as n goes to infinity of sin(x/n) / cos(x/n) so the limited development of sin (X) when X goes to 0 is sin(X) = X + eps(X) where eps is a function that goes to 0. and the limited development of cos(X) when X goes to 0 is cos(X) = 1 + eps(X) so we have tan(x/n) = sin(x/n)/(cos(x/n) = ( x/n + eps(x/n) ) / ( 1 + eps(x/n) ) -> ( x/n) / 1 = x/n I apology for my english, hope that still understandable...
Isn't y=cx, where c is a constant and sin(c)=0 (meaning c is equal to n*pi for some integer n) also a solution? Was this solution lost when dividing by sin^2(u) without checking the cases where it was zero?
Hi Sam, thanks so much for wanting to support me and liking my channel. However, I do not have a patreon account. You can simply keep supporting me by watching my videos and sharing them with others. Greatly appreciate your kindness! bprp
Dalci Diogo Let's say you put constants after integration on both sides... C1 and C2. After integration you could just transpose C1 to the right side so the constant is C2-C1 on the right side and no constant on the left side. But C2-C1 is still a constant. So we just simplify it and write it as C.
Can you do a video on how to solve any integral( i mean like what to do when these type of integral comes like splitting the integral, when to do u substitution)
Ben Dova We have: lim n->inf. (1/n^(1/n))=lim n->inf. (1/n)^(1/n). Use substitution x=1/n. When n goes to infinity, x goes to 0+ (since 1/n is a positive number because n is approaching positive infinity, the positive number 1 divided by another positive number is also positive). So we have lim n->inf. (1/n)^(1/n)=lim x->0+ (x^x)=1. lim x->0+ (x^x) is approaching 1 if x is approaching 0 from the positive side of x. So lim n->inf. (1/n^(1/n))=1.
Thank you so much for the awesome video, i recently had a calc 3 test and i was wondering if you could help me with the answer to one question: A particle moves with the position function r(t)=(8cos(t),8sen(t),8t) Fin the normal normal component of acceleration You think you could help me? Zorry is it doesn't make sense, its in spanish and i roughly translated it myself
Did you asked your girlfriend to make a full cover of the outro song? Also love you differential equations videos, i really struggle with it a lot in college but your videos have really helped me, could you try to make more multivariable calculus videos?
Hey blackpenredpen, your thumbnail shows the wrong question from the one you solved in the video, the thumbnail one doesnt have the x term in the -sin^2 (y/x) part at the end
Impressive but I do have a question. Isn't cot^(-1)(c) just a constant? It doesn't really matter if you put it in or out it's still a constant. I thought it doesn't matter but correct me if I'm wrong.
I think it is Bernoulli equation and can be easily reduced to linear x^2 (xdx+ydy) + 2y(xdy-ydx)=0 Lets expand and group it x^3dx+x^2ydy+2yxdy-2y^2dx=0 (x^3-2y^2)dx+(x^2y+2yx)dy=0 x^3-2y^2+(x^2+2x)ydy/dx=0 This equation can be written in the form dy/dx+P(x)y=Q(x)y^{r} but this is not neccessary Let substitute u=y^2 and then use variation of parameter or integrating factor to solve linear equation x^3-2y^2+(x^2+2x)ydy/dx=0 2x^3-4y^2+(x^2+2x)2ydy/dx=0 u=y^2 u=2ydy/dx 2x^3-4u+(x^2+2x)du/dx=0 du/dx-4/(x^2+2x)u=-2x^3/(x^2+2x) It is also homogeneous in the form d/dx y(x)=y(x)/x+g(x)f(y(x)/x) and thats why substitution for homogeneous equation works Integrating factor of one variable also exists for this equation Bernoulli equation in the form dy/dx+P(x)y=Q(x)y^{r} has separable integrating factor
@Alien Your solution is nice and probably expected I found this form of homogeneous equation only in Maple documentation because scripts in my native language give only dy/dx=f(y/x) as homogeneous and that's why i firstly recognize this equation as Bernoulli
I found equation for you (3xy+y^2)+(3xy+y^4)dy/dx=0 If we find nonzero functions such that yF(x+y)-3xG(x+y^3/3)=0 we can find integrating factor or it is another way to solve it
Try this one (1-x^2)d^2y/dx^2-xdy/dx+9y=0 This equation can be solved by guessing particular solution and reduction of order but shows that inverse trig substitutions are not always the fastest one By the way polynomial is particular solution to the equation i gave The integral which you will get I would calculate using Euler subsitution because in my opinion is faster than inverse trig substitution Actually i gave this equation for you to show that Yes you use inverse trig substitution to get rid of radical An example of trig substitution is Weierstrass substiutution Do you see the difference ?
You didn't see mistake ? your will not work Try to solve it as i suggested and you will get integral which shows that Euler substitution can be faster then inverse trig substitution You probably didn't try to solve it in that way (guessing polynomial solution and reduction of order)
I could record video but my English is not very well and I have limited time on youtube I know he is Chinese but i think that he at least lives in English spoken country
Wich interval are you working on? How can you be sure you haven't divided by zero when you divided by X. Moreover you're not able to use inverse trigo function if you don't check the interval
Second will be also Bernoulli after substitution y = 1/2 (x^2-u) (x^2-u)(x^2-1/2(x^2-u)+x)+(x^2-(x^2-u))1/2(2x+du/dx)=0 (x^2-u)(1/2x^2+x+1/2u)+1/2u(2x+du/dx)=0 1/2(x^2-u)(x^2+2x+u)+xu+1/2udu/dx = 0 1/2(x^4+2x^3+x^2u-ux^2-2xu-u^2)+xu+1/2udu/dx = 0 1/2*x^4+x^3-xu-1/2u^2+xu+ 1/2udu/dx=0 x^4+2x^3-u^2+udu/dx=0 udu/dx-u^2=-(x^4+2x^3) du/dx - u=-(x^4+2x^3)*1/u and that is Bernoulli equation
Pedro Lima It's not 2iπ = 0, it's ln(1) = 2πni where n is an integer ≥ 0. This doesn't imply that 2πi = 0, only that ln(1) has multiple values just like √(4) = ±2
natan9065 the log function is only defined in {z€C / z=/=0 and O=/=pi}. Note that I'm using € as the "belongs to" simbol and O as theta (to signify angles). Although, if memory is working well (which it probably isn't), you can define it for O=pi. But you'll lose some other angle.
![Seungmin "그렇게, 천천히, 우리(As we are)" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/kAzmhLHePqU/mqdefault.jpg)
Thank you Thank you Thank you Thank you So Very Much Sir..Love you to sir. Really you are great sir.
Ved Prakash my pleasure to help! Hope you like it!
Why not! Sir.... You are my best teacher of mathematics. Really, I will be pray to god that god give you long life. 🤗🤗🤗👍👍👍👍👍👍🤝🤝🤝
Ved Prakash thank you thank you!
thanks for giving us a nice problem to solve too!
Ved Prakash bhai sach kahoon toh Yeh itna mushkil nahi hai
bruh its a silent night here, me alone in my room and that piano music all of a sudden at the scared the shit out of me. legit.
I love this guy so much
Good explanatin sir.
Keep doing this good work for Students.
Hello. First of all great content on your channel, love it and keep it up! :) Secondly I am in the first year of my master degree, mechanical engineering. For some reason in all these years of maths classes we have never actually talked about Secant and Cosecant. I know about cos, sin, tan and cot. So my idea for next video: do a quick rundown for sec and cosec, basic definiton, usefulness etc? I guess I could google all that, but I just enjoy watching you explaining stuff since it's far more interesting. Good luck and thanks :)
nice solution, nice piano at the end
That solution was a work of art :-) You're so good at presenting these!
Yes, this is a great video. Thank you! You should post more differential equation videos, too. You really do help with problem solving skills. Excellent.
I never took differential equations but I’m so fascinated by these video. I took up to call 3 and linear algebra but de isn’t required for my major
I did it in my ninth grade.
You could also get y=x*arctan(1/(ln(abs(x))+c))
And y = x*arccot(ln|x| + C)
Sir please make videos on Olympiad problems
The video’s thumbnail forgot the x next to the sin^2. Missing an x there.
Alan Agnew ah!! Thanks for pointing that out
Alan Agnew yes.... But it was solvable too
Ranit Chatterjee how do me that
Maity tell me how I show you....!
Ranit Chatterjee just right it down ... Here in comment section ...
If you think it's not possible then left
Hey. Can you do a proof that the limit as n goes to infinity of tan(x/n) is equal to x/n?
If you know the limited development, this is quite easy:
as you know tan (X) = sin(X)/cos(X)
so we are looking for the limit as n goes to infinity of sin(x/n) / cos(x/n)
so the limited development of sin (X) when X goes to 0 is sin(X) = X + eps(X) where eps is a function that goes to 0.
and the limited development of cos(X) when X goes to 0 is cos(X) = 1 + eps(X)
so we have tan(x/n) = sin(x/n)/(cos(x/n) = ( x/n + eps(x/n) ) / ( 1 + eps(x/n) ) -> ( x/n) / 1 = x/n
I apology for my english, hope that still understandable...
Could you explain the change to polar coordinates in integration?
Edit: 100k HYPE!!
Could you make a video on the epsilon delta definition of limits please as you're the best at making us understand maths.
Isn't y=cx, where c is a constant and sin(c)=0 (meaning c is equal to n*pi for some integer n) also a solution? Was this solution lost when dividing by sin^2(u) without checking the cases where it was zero?
Thank you so much
You have no idea how you're great thank uuuuu so much for your help 🌷❤
Do you a patreon account ? I'd love to support you and the great content you are providing :)
Hi Sam, thanks so much for wanting to support me and liking my channel. However, I do not have a patreon account. You can simply keep supporting me by watching my videos and sharing them with others. Greatly appreciate your kindness!
bprp
I've just 1 question sir:
Why did you put the constant C on the right limb only?
Dalci Diogo Let's say you put constants after integration on both sides... C1 and C2. After integration you could just transpose C1 to the right side so the constant is C2-C1 on the right side and no constant on the left side. But C2-C1 is still a constant. So we just simplify it and write it as C.
Reeta Singh Ok, now I understand. Thanks so much
Can you do a video on how to solve any integral( i mean like what to do when these type of integral comes like splitting the integral, when to do u substitution)
A rookie question: why can we separate du/dx as du and dx? d/dx() is one operator isn't it? I mean it looks logical, but what's the math behind it?
Probably because dx, dy, du, dw, whatever expression, are infinitesimal number. I would like to know a better explain too
You should do a video about laplace transformation.
More!!
Cotangent isn’t one-to-one. When you apply inverse cotangent, do you need a k*pi anywhere to correct for that?
What is fourier transform
Hey can you do a video on how to find the limit as n goes to infinity of: 1/(n^(1/n)) ? ?
Ben Dova We have:
lim n->inf. (1/n^(1/n))=lim n->inf. (1/n)^(1/n). Use substitution x=1/n. When n goes to infinity, x goes to 0+ (since 1/n is a positive number because n is approaching positive infinity, the positive number 1 divided by another positive number is also positive). So we have
lim n->inf. (1/n)^(1/n)=lim x->0+ (x^x)=1. lim x->0+ (x^x) is approaching 1 if x is approaching 0 from the positive side of x.
So lim n->inf. (1/n^(1/n))=1.
would it work to then write it as y = x arccot(lnx + C)?
Great. Video!
Sir please do a video on laplace transform... thanks
3lwii müsiic i have them already. Check my playlists
Also, I think 3blue1browns next video is going to be on Laplace transforms
5:48
This is how King Crimson works
great job !
How can you make "u" an angle (inside the sin) as well as normal number. I thought you couldn't do that
u isn't really being an angle in this case, its just another number.
Horep Alright, thanks
Music at 7:07 ? 老師！
Could you bring sin(y/x)^2 to the left hand side on line 2 and then do integrating factor method?
No, because sin is a function of y/x and not x.
Thank you so much for the awesome video, i recently had a calc 3 test and i was wondering if you could help me with the answer to one question:
A particle moves with the position function r(t)=(8cos(t),8sen(t),8t)
Fin the normal normal component of acceleration
You think you could help me? Zorry is it doesn't make sense, its in spanish and i roughly translated it myself
Sir I have a question for u.If e^(x*i)=x Find x
Plz do something cool for 100k special !!
Like idk , livestream of solving integrals togetger would be fun I guess , up yo you!
Did you asked your girlfriend to make a full cover of the outro song? Also love you differential equations videos, i really struggle with it a lot in college but your videos have really helped me, could you try to make more multivariable calculus videos?
Sir that -sin^2x will become positive as after going to other side it will add in it
We were dividing by it
blackpenredpen okk. Thank you
Sir please
Can you to solution
int e^cosx (sin (sinx))dx
Shouldn’t there also be a +pi*n at the very end where n is an integer
No actually a +x*n*pi
Give a try to this:
I= int(x^x - x^(-x))dx
from 0 to 1
Hey blackpenredpen, your thumbnail shows the wrong question from the one you solved in the video, the thumbnail one doesnt have the x term in the -sin^2 (y/x) part at the end
Thanks!!! I just fixed that!
Impressive but I do have a question. Isn't cot^(-1)(c) just a constant? It doesn't really matter if you put it in or out it's still a constant. I thought it doesn't matter but correct me if I'm wrong.
I'll just say it mathematically. cot^-1(a+b) is not the same as
cot^-1(a)+cot^-1(b).
can you do a video of integral ( 1/(x+1)! )
Sir please can you do my DE problem ... I tried 2 hour ...But failed ..
x^2 (xdx+ydy) + 2y(xdy-ydx)=0
Alien in disguise but then ... How to proceed after that ... Even if I assume now x/y=z then ... Nothing happens
Alien in disguise btw your rearrangement was great
I think it is Bernoulli equation and can be easily reduced to linear
x^2 (xdx+ydy) + 2y(xdy-ydx)=0
Lets expand and group it
x^3dx+x^2ydy+2yxdy-2y^2dx=0
(x^3-2y^2)dx+(x^2y+2yx)dy=0
x^3-2y^2+(x^2+2x)ydy/dx=0
This equation can be written in the form
dy/dx+P(x)y=Q(x)y^{r}
but this is not neccessary
Let substitute u=y^2 and then use variation of parameter or integrating factor to solve linear equation
x^3-2y^2+(x^2+2x)ydy/dx=0
2x^3-4y^2+(x^2+2x)2ydy/dx=0
u=y^2
u=2ydy/dx
2x^3-4u+(x^2+2x)du/dx=0
du/dx-4/(x^2+2x)u=-2x^3/(x^2+2x)
It is also homogeneous in the form
d/dx y(x)=y(x)/x+g(x)f(y(x)/x) and thats why substitution for homogeneous equation works
Integrating factor of one variable also exists for this equation
Bernoulli equation in the form dy/dx+P(x)y=Q(x)y^{r}
has separable integrating factor
@Alien Your solution is nice and probably expected I found this form of homogeneous equation only in Maple documentation because scripts in my native language give only dy/dx=f(y/x) as homogeneous and that's why i firstly recognize this equation as Bernoulli
I found equation for you
(3xy+y^2)+(3xy+y^4)dy/dx=0
If we find nonzero functions such that
yF(x+y)-3xG(x+y^3/3)=0 we can find integrating factor
or it is another way to solve it
I don't know why he doesn't have so many likes
Please solve this
Integral sqrt(x^4+1)/(x^2+1)^2
Atleast prove it cannot be done...i will be happy
There is missing solution: -sin^2 (u)=0, then u=pi*k, when k is real. Then y=x*pi*k is also a solution
I am pretty sure. Everyone here loves maths.
I have a question:
cot(x)=1/tan(x).
cot^-1 (x) = ( 1/tan(x) )^-1
cot^-1 (x) = tan(x)
why didn't you/he do that?
Jakob R, ow wow! thanks :)
Don't forget the synthesis ... So you can solve it in R
(dy/dx) = (4x+y+1)²
Can you please solve this sir
4x+y=z diff w.r.t x
Alien in disguise thanks, I just missed that key element.
d²(y)/d(x)²= -(sin(y)/3π)
I'm just curious ...
Nice ! :)
You don't like the "prime" notation for derivatives?
in diff eq, I prefer either dy/dx or the D notation.
blackpenredpen I would love too see you using the D notation like Dr Peyam
Sir please help 1/(x+e^x) & (cos(x^2))/(x^2)
In case calculator has no cot-1(x) use tan-1(1/x)..?
yes
Try this one
(1-x^2)d^2y/dx^2-xdy/dx+9y=0
This equation can be solved by guessing particular solution and reduction of order
but shows that inverse trig substitutions are not always the fastest one
By the way polynomial is particular solution to the equation i gave
The integral which you will get I would calculate using Euler subsitution
because in my opinion is faster than inverse trig substitution
Actually i gave this equation for you to show that
Yes you use inverse trig substitution to get rid of radical
An example of trig substitution is Weierstrass substiutution
Do you see the difference ?
You didn't see mistake ? your will not work Try to solve it as i suggested and you
will get integral which shows that Euler substitution can be faster then inverse trig substitution
You probably didn't try to solve it in that way (guessing polynomial solution and reduction of order)
I could record video but my English is not very well and I have limited time on youtube
I know he is Chinese but i think that he at least lives in English spoken country
希望你能證明isinθ+cosθ=e^iθ！
我在i^i那部卡在這裡😢
Please help me solve that: 10^x=x+100
Amazing! and dy/dx=k* y* ln(a/y) please can you help me?
If a and k are constants you have separable , problem with integral ?
yeah... the explicit solution is y=a*e^((e^-kx)/A) and A=e^C ????
More
Help: integrate sin(ax)sin(bx)sin(cx)
Sir please help me in this
Prove that
Sin(2π÷7) +Sin(4π÷7) +Sin(8π÷7) =√7÷2
n=a+bi, f(a+bi)=b+ai, f(n)=?
The x is missing in intro
😮
Wich interval are you working on? How can you be sure you haven't divided by zero when you divided by X. Moreover you're not able to use inverse trigo function if you don't check the interval
i have a D.E. easy (4yx^3)dx+(1+x^4)dy=0 IT`s AWESOM.
but, this is not easy 2y(x^2-y+x)dx+(x^2-2y)dy=0
First D.E. is separable
For the second exists integrating factor of one variable
Multiply second equation by e^{2x} and you will get exact equation
Yes, the second multiplied by e ^ (2x) the whole equation.
Second will be also Bernoulli after substitution y = 1/2 (x^2-u)
(x^2-u)(x^2-1/2(x^2-u)+x)+(x^2-(x^2-u))1/2(2x+du/dx)=0
(x^2-u)(1/2x^2+x+1/2u)+1/2u(2x+du/dx)=0
1/2(x^2-u)(x^2+2x+u)+xu+1/2udu/dx = 0
1/2(x^4+2x^3+x^2u-ux^2-2xu-u^2)+xu+1/2udu/dx = 0
1/2*x^4+x^3-xu-1/2u^2+xu+ 1/2udu/dx=0
x^4+2x^3-u^2+udu/dx=0
udu/dx-u^2=-(x^4+2x^3)
du/dx - u=-(x^4+2x^3)*1/u
and that is Bernoulli equation
See, its always U! Hahahahahahahahahaha
do you forget the minus?
Flixx K. Nope he is right. Integral of -csc^2 (u) du is cot (u).
ok, thx!
Ur welcome. :)
But... this is not homogeneous, this is an exact differential equation
Why is everyone calling him "Sir"?
Because he deserves to be
Why not?
Sir, it will be my pleasure if you solve this problem for me,,
|2x/(x-2)|
Easy 😁
Typo in thumbnail
ISI PISI!
have a completely false solution for fun:
dy/dx = y/x - sin²(y/x)
d/d * y/x = y/x - sin²(y/x)
y/x = y/x - sin²(y/x)
0 = - sin²(y/x)
y = 0; x ≠ 0 OR y/x = π*k
If ln(-1)=iπ then I ln(1)=2iπ=0 then i=0. Where is the error?
Pedro Lima It's not 2iπ = 0, it's ln(1) = 2πni where n is an integer ≥ 0. This doesn't imply that 2πi = 0, only that ln(1) has multiple values just like √(4) = ±2
natan9065 Thanks! But why can't n be
Pedro Lima I don't see why not, negative just means rotating the opposite direction! Thanks
natan9065 the log function is only defined in {z€C / z=/=0 and O=/=pi}. Note that I'm using € as the "belongs to" simbol and O as theta (to signify angles).
Although, if memory is working well (which it probably isn't), you can define it for O=pi. But you'll lose some other angle.