solving the differential equation y''+y=tan(t) by variation of parameters

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  • เผยแพร่เมื่อ 9 พ.ย. 2024

ความคิดเห็น • 110

  • @raotalha6076
    @raotalha6076 7 ปีที่แล้ว +29

    my teacher solved it in more easy way

    • @therodeo4372
      @therodeo4372 7 ปีที่แล้ว +153

      DONT DISRESPECT THE MAN HES FUCKING AMAZING
      KEEP ON DOING YOUR THING BPRP

    • @koober_
      @koober_ 7 ปีที่แล้ว +22

      Yes, there are easier ways, such as just using the formula, but this way (as well as the method involving matrices) kind of gives you some background on how to derive the formulas.

    • @poojarani6046
      @poojarani6046 6 ปีที่แล้ว +25

      So why are you watching dis....
      Go n study to your teacher......dont show off man!

    • @ThilebanTheEngineer
      @ThilebanTheEngineer 6 ปีที่แล้ว

      Watch my channel for more easy way..

    • @Ethan-rw7ub
      @Ethan-rw7ub 6 ปีที่แล้ว +9

      Hey guys chill alright? It's completely normal that there are faster ways to solve a question, don't get so butthurt.

  • @tagekumarpaul7474
    @tagekumarpaul7474 4 ปีที่แล้ว +17

    For anyone watching, after he gets the system of equations that he has to solve, you can use cramer's rule to get v1' and v2' and then integrate to get v1 and v2. It's a slightly faster way. Great video though. I love this guys channel.

  • @dessa8242
    @dessa8242 3 ปีที่แล้ว +1

    you are a lifesaver guy!!! I have been hovering multiple videos about this topic, none of the videos had any clear and confidential explanation, rather than yours. they just be there for the purpose of collecting subscriptions.

  • @KeyMan137
    @KeyMan137 7 ปีที่แล้ว +23

    For those who want the final solution: y = C1 cos(t) + C2 sin(t) - cos(t) ln|sec(t) + tan(t)|

    • @blackpenredpen
      @blackpenredpen  7 ปีที่แล้ว +5

      Spencer Key yay!

    • @carultch
      @carultch ปีที่แล้ว

      Problem: find the Fourier Series.

  • @jaimeduncan6167
    @jaimeduncan6167 7 ปีที่แล้ว +7

    Excellent work as always. By the way it was easier to substitute v2 derivative instead in the first equation, exploiting the zero.

  • @kindofhungry
    @kindofhungry 7 ปีที่แล้ว +7

    Can you do one using wronskians and integral? Ex: u_1 = int((y_2(t)*g(t))/W(y_1,y_2))

  • @helldogforever
    @helldogforever 5 ปีที่แล้ว +2

    I understand this now with 1000% clarity.

  • @jevcampbell2301
    @jevcampbell2301 3 ปีที่แล้ว +3

    you my friend deserve a medal. You will reach 1 mil subs

    • @GSHAPIROY
      @GSHAPIROY ปีที่แล้ว

      It happened, on November 11, 2022.

  • @eess2396
    @eess2396 5 หลายเดือนก่อน

    太谢谢你了,我考前复习正好看到这个问题不懂,你这个办法也挺好的,用包含矩阵的积分公式似乎对我来说难以记忆,如果真忘了那我只好推导了哈哈老师说我们时间很够。

  • @therodeo4372
    @therodeo4372 7 ปีที่แล้ว +1

    THANK YOU!!!! PLEASE KEEP IT COMING I HAVE CALC 3 IN THE FALL AND LINEAR ALGEBRA IN THE SPRING

  • @reubenwilliammpembe667
    @reubenwilliammpembe667 6 ปีที่แล้ว +3

    you are the best!!!!
    #RespectFromSouthAfrica

  • @jackkalver4644
    @jackkalver4644 ปีที่แล้ว +1

    Is variation of parameters ever easier than linear integration? Here’s the same problem solved directly:
    y”+y=tan t
    y”*cos t+y*cos t=sin t
    y’*cos t+y*sin t=C1-cos t
    y’*sec t+y*tan t*sec t=C1*sec^2 t-sec t
    y*sec t=C1*tan t-ln|sec t+tan t|+C2
    y=C1*sin t-cos t*ln|sec t+tan t|+C2*cos t

  • @osunrinadebukola6082
    @osunrinadebukola6082 6 หลายเดือนก่อน +1

    Very helpful. God bless you

  • @bird9
    @bird9 3 ปีที่แล้ว

    THANK YOU BlackPenRP

  • @iindifferent
    @iindifferent 7 ปีที่แล้ว +2

    Excellent video! Thank you so much.

  • @blackpenredpen
    @blackpenredpen  7 ปีที่แล้ว +11

    y"+y=tan(t)

    • @dyer308
      @dyer308 7 ปีที่แล้ว +1

      I love all your videos!

    • @batubulgur
      @batubulgur 7 ปีที่แล้ว +1

      blackpenredpen blue pen? there is something wrong with this video... :)

    • @blackpenredpen
      @blackpenredpen  7 ปีที่แล้ว +1

      raees khan thanks!!

  • @Physticuffs
    @Physticuffs 7 หลายเดือนก่อน

    Finally i understand this. Thank you!

  • @valentinlishkov9540
    @valentinlishkov9540 6 หลายเดือนก่อน

    Issue:
    What is a differential of an irrational argument?
    Let a= some rational approximation, and A be the irrational number itself (if that makes sense).
    Then A - a > dA and there is no way a + dA > A

  • @hmjawad087
    @hmjawad087 4 ปีที่แล้ว +1

    Just Amazing Brother..!!!

  • @ssdass4303
    @ssdass4303 ปีที่แล้ว +2

    why dont u just plug in V2' which u just solved

  • @cassiechandler4314
    @cassiechandler4314 7 ปีที่แล้ว +1

    Very Helpful

  • @Ggon636
    @Ggon636 4 ปีที่แล้ว +5

    After finding V'2 why can we not plug it into one of the equations to acquire V'1??

    • @aritro3
      @aritro3 4 ปีที่แล้ว +1

      you can

    • @tomatrix7525
      @tomatrix7525 3 ปีที่แล้ว +1

      Yep, you can get it many ways. Whatever floats your equation bro

  • @tobechukwublessed4274
    @tobechukwublessed4274 ปีที่แล้ว

    Thanks you

  • @poojarani6046
    @poojarani6046 6 ปีที่แล้ว +1

    You and your methods are amazing and easily understandable also....

  • @johansvensson830
    @johansvensson830 6 ปีที่แล้ว +1

    Thank you so much!

  • @raptokvortex
    @raptokvortex 7 ปีที่แล้ว +1

    Cool. We've only learnt how to solve linear second order differential equations by inspection so far.

  • @borg972
    @borg972 6 ปีที่แล้ว +2

    Thanks! would love to see more examples for variation of parameters!

  • @jrcano1889
    @jrcano1889 6 ปีที่แล้ว

    Thank you, my teacher want us to do it this way and not the wronskian method

  • @hanwadou1777
    @hanwadou1777 7 ปีที่แล้ว

    Good job I love it so much

  • @jameshenner5831
    @jameshenner5831 6 ปีที่แล้ว +1

    plotting this solution reveals a weirdly lumpy periodic function with the period of 2*pi. The lumpiness is kind of interesting if the coefficients of sine and cosine are relatively small. picture: i.imgur.com/Pnu2Yua.jpg

  • @rohansawai6534
    @rohansawai6534 4 ปีที่แล้ว

    #RespectfromIndia

  • @sunnymanish5514
    @sunnymanish5514 ปีที่แล้ว

    6:18 - 6:22 was epic

  • @zainabalmusawi7749
    @zainabalmusawi7749 6 ปีที่แล้ว +2

    طريقة الحل جدا مفهومه شكرا

  • @fernandogallardo3477
    @fernandogallardo3477 6 ปีที่แล้ว +15

    I have to do it via wronskian

    • @skepticmoderate5790
      @skepticmoderate5790 4 ปีที่แล้ว

      Have to? That's the easier way. I always prefer a formula.

    • @benedictmaddara9248
      @benedictmaddara9248 4 ปีที่แล้ว

      This is the safest solution possible. So, why not? haha

  • @charlesrothauser1328
    @charlesrothauser1328 4 ปีที่แล้ว

    Where is C from the particular solution in the final answer?

  • @sirius.aeternus
    @sirius.aeternus 3 ปีที่แล้ว +1

    he makes me laugh when he laughs lol

  • @isaacdesantigoisaac1319
    @isaacdesantigoisaac1319 7 ปีที่แล้ว

    i love your videos

  • @joyceadeniyi9005
    @joyceadeniyi9005 6 ปีที่แล้ว

    Thanks so much

  • @mkt92z
    @mkt92z 5 ปีที่แล้ว +1

    Is it easier to use wronskian couple with cramers rule?

  • @gulaykeskin8647
    @gulaykeskin8647 6 ปีที่แล้ว

    thank you

  • @tanyaradzwakufa8709
    @tanyaradzwakufa8709 3 ปีที่แล้ว

    why not use the wroskian approach

  • @carlmarionmanlapaz4697
    @carlmarionmanlapaz4697 5 ปีที่แล้ว +2

    hello there how to do that when i have more than 3 equations? are there techniques to quicken the solving?
    @blackpenredpen

  • @arkansh.h.1313
    @arkansh.h.1313 5 ปีที่แล้ว

    hi thnx a lot. And how about this diff. eq. y''+5y'+11=tan(x)?

  • @susovanmanna8323
    @susovanmanna8323 2 ปีที่แล้ว

    🙏🙏sir u r amazing

  • @iradnuriel9087
    @iradnuriel9087 7 ปีที่แล้ว

    Hi blackpenredpen , I didn't realize why can I find the harmonic solution by solving a polinom? (sorry if my English is incorrect, I don't good in English)

  • @comic4u001
    @comic4u001 4 ปีที่แล้ว

    Very helpfull. How Can i solve this equation please? y''-2y'-3y=3sinh(2x)-12cosh(2x)

    • @carultch
      @carultch ปีที่แล้ว

      Given:
      y" - 2*y' - 3*y = 3*sinh(2*x) - 12*cosh(2*x)
      For this example, I'd use the Laplace transform, and assign arbitrary initial conditions. This can also work using the method of undetermined coefficients, because sinh(2*x) and cosh(2*x) are both a linear combinations of e^(-2*x) and e^(2*x), so you can assume both of these as your ansatz of the particular solution for undetermined coefs.
      Let y(0) be u, and let y'(0) be v. Also let Y(s) = £{y(x)}
      £{y"} = s^2*Y(s) - v - u*s
      £{-2*y'} = -2*s*Y(s) + 2*u
      £{-3*y} = -3*Y(s)
      £{3*sinh(2*x))} = 6/(s^2 - 4)
      £{12*cosh(2*x))} = 12*s/(s^2 - 4)
      Thus:
      (s^2 - 2*s - 3)*Y(s) - v - u*s + 2*u = 6/(s^2 - 4) - 12*s/(s^2 - 4)
      Shuffle initial conditions to the right:
      (s^2 - 2*s - 3)*Y(s) = 6/(s^2 - 4) - 12*s/(s^2 - 4) + v + (s - 2)*u
      Change right side to common denominators:
      (s^2 - 2*s - 3)*Y(s) = (6 - 12*s + (u*s - 2*u + v)*(s^2 - 4))/(s^2 - 4)
      Isolate Y(s):
      Y(s) = (6 - 12*s + (u*s - 2*u + v)*(s^2 - 4))/((s^2 - 4)*(s^2 - 2*s - 3))
      Factor denominator and expand/gather numerator:
      Y(s) = (u*s^3 + (v - 2*u)*s^2 - (12 + 4*u)*s + 8 u - 4 v + 6)/((s - 2)*(s + 2)*(s + 1)*(s - 3))
      Partial fractions:
      Y(s) = A/(s - 3) + B/(s + 1) + C/(s - 2) + D/(s + 2)
      From experience, we know that the two solutions whose denominators are factors of the characteristic polynomial in front of Y(s), they will be the two solutions whose output depends on initial conditions. As such , C and D are the two coefficients that will be from the particular solution, and won't depend on initial conditions. Heaviside cover-up works great for finding them. So we might as well just leave A and B alone, as unknowns for the general solution.
      C = (u*2^3 + (v - 2*u)*2^2 - (4*u + 12)*2 + 8*u - 4*v + 6)/((2 +2)*(2 + 1)*(2 - 3)) = 3/2
      D = (u*(-2)^3 + (v - 2*u)*(-2)^2 - (4*u + 12)*(-2) + 8*u - 4*v + 6)/((4 + 4)*(4 + 1)*(4 - 3)) = -3/2
      Y(s) = A/(s - 3) + B/(s + 1) + 3/2/(s - 2) - 3/2/(s + 2)
      Take inverse Laplace:
      y(x) = A*e^(3*x) + B*e^(-x) + 3/2*e^(2*x) - 3/2*e^(-x)
      And we can consolidate our particular part of the solution as a single sinh:
      y(x) = A*e^(3*x) + B*e^(-x) + 3*sinh(2*x)

  • @ΘέκλαΧατζηκώστα
    @ΘέκλαΧατζηκώστα 6 ปีที่แล้ว

    y''+4y=cos(2x)

    • @yolcu8609
      @yolcu8609 6 ปีที่แล้ว +1

      You can use undetermined coefficient method

  • @begadanan2649
    @begadanan2649 11 หลายเดือนก่อน

    just use wronskian instead and easier and workds

  • @GHOSTrex1324
    @GHOSTrex1324 4 ปีที่แล้ว

    lit outro music

  • @nhaminh5717
    @nhaminh5717 3 ปีที่แล้ว

    What about y''+y'=tan(x)

  • @yahiazakarialadhem9411
    @yahiazakarialadhem9411 5 ปีที่แล้ว

    thank's

  • @WilliamBlake-yj2yu
    @WilliamBlake-yj2yu 2 ปีที่แล้ว

    is this still cauchy euler method applied?

  • @daniell321
    @daniell321 6 ปีที่แล้ว

    taking this adv ODE in year 3 mechanical engineering, will take adv PDE next year hope u can cover the topics in adv PDE love

  • @Rundas69420
    @Rundas69420 7 ปีที่แล้ว

    We can all be happy that the original question is y"+y=tan(t) and not y"-y=tan(t).

    • @blackpenredpen
      @blackpenredpen  7 ปีที่แล้ว

      That is correct!

    • @leroyj9044
      @leroyj9044 6 ปีที่แล้ว

      I'm struggling along with y''-y=sinh(2x)

  • @BukhalovAV
    @BukhalovAV 7 ปีที่แล้ว

    Why there is no the coefficient C_3 at cos(t)*ln(abs(sec(t)+tan(t))) in the answer?

  • @programmingpython584
    @programmingpython584 3 ปีที่แล้ว

    I am not getting how he takes y1=cost and y2 = sint. Please explain that

  • @matheusmodesto5794
    @matheusmodesto5794 5 ปีที่แล้ว

    Why y1' +y2' = f(t)/a is valid? Is there have any theorems?

    • @repvoo2399
      @repvoo2399 11 หลายเดือนก่อน

      you can watch the previous video about the variation of parameters, introduction, and idea, in case you are still alive.

  • @burningoyster
    @burningoyster 7 ปีที่แล้ว

    In what video do you explain the y_p equation?

  • @nurfitrah6126
    @nurfitrah6126 7 ปีที่แล้ว

    can you help me how to solve y''+y= sec (theta) tan (theta). im confused when it has (theta)

  • @yarsak8067
    @yarsak8067 5 ปีที่แล้ว +1

    Great vid as always! By the way, would this method also work with higher order derivatives (f.ex. y''', y"" etc)? Thanks!

  • @taimooralibukhari
    @taimooralibukhari 6 ปีที่แล้ว

    if sin^2t+cos^2t=1 then v1'+v1=2v1'? isnt it?

  • @shanaraj6580
    @shanaraj6580 2 ปีที่แล้ว

    👏🏽👏🏽👏🏽👏🏽👏🏽

  • @nurshafiqahjamaludin1702
    @nurshafiqahjamaludin1702 5 ปีที่แล้ว

    ive just got the y1 for the yh, how i can solve it???

  • @Kay-dx8vm
    @Kay-dx8vm 6 ปีที่แล้ว

    My equation is a 3rd order nonhomogeneous. How can I solve it ?

  • @husamrajeh179
    @husamrajeh179 6 ปีที่แล้ว

    You forgot to write y2v2 its sinx(-cosx)

    • @blackpenredpen
      @blackpenredpen  6 ปีที่แล้ว

      Day orange6 ?

    • @stumbling
      @stumbling 6 ปีที่แล้ว

      He did, it cancelled out. :)

  • @alwinpriven2400
    @alwinpriven2400 7 ปีที่แล้ว

    what's y'' is it the second derivative of y?

  • @isharauditha4257
    @isharauditha4257 3 ปีที่แล้ว

    how do you become this much smart

  • @Malikasoy5828
    @Malikasoy5828 3 ปีที่แล้ว

    한국인처럼 생겼군요 ...... 김태형 아시 잖아요

  • @saidinesrine3689
    @saidinesrine3689 5 ปีที่แล้ว

    what is the sect ???? THANKS

    • @carultch
      @carultch ปีที่แล้ว

      sec(t) = 1/cos(t)

  • @Ocklepod
    @Ocklepod 7 ปีที่แล้ว

    Not sure if title of video is quotation marks or 2 of '

  • @tobechukwublessed4274
    @tobechukwublessed4274 ปีที่แล้ว

    Cooll

  • @dirtybee224
    @dirtybee224 6 ปีที่แล้ว +1

    My teacher makes use of matrix to make it look easier and faster.

  • @takudzwabveke2599
    @takudzwabveke2599 7 ปีที่แล้ว

    wakanyanya

  • @husamrajeh179
    @husamrajeh179 6 ปีที่แล้ว

    At the end u forgot to ad y2u2

  • @hecadream9717
    @hecadream9717 3 ปีที่แล้ว

    uau

  • @slaywithkeke7129
    @slaywithkeke7129 3 ปีที่แล้ว

    Thank you very much. Much appreciated.

  • @justin-ls5yc
    @justin-ls5yc 3 ปีที่แล้ว

    thank you