undetermined coefficients, diff eq, sect4.5#19

[@nahdoodavis3924]

You just helped me so much on my ODEs test. I wish my professor taught the way you did! Great job.

[@blackpenredpen]

You're welcome! I hope you did well on the exam!

[@Sleepy46169]

Dear sir. Your final answer of the Y sub h is C sub 1e^t + C sub 2e^2t. Is it alright that the final answer of the Y sub h, its roots were shuffled? Like mine is C sub 1e^2t + C sub 2e^t. Is this the same?

[@hugopurpeq]

@@Sleepy46169 yes its the same

[@boing7771000]

I love how enthusiastic and natural this guy is:)

[@betechtechnology942]

thank you very much you the best good techniques especially on 2nd derivative it could have been very very long

[@--8251]

ngl, I was too worried about the thermal detonator in his hand to focus on the problem

[@hans3331000]

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[@kabirkohli9920]

You're literally the reason I'm gonna pass my exam

[@ianmoseley9910]

2B or negative 2B, that is the question ...

[@dylongoodyear7718]

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[@nsifrahman9031]

@@dylongoodyear7718 Can I enjoy you?

[@mcleanephatha]

@@nsifrahman9031 You can enjoy me!

[@hellmuth26]

every time I have a 2B i say "or not to be," it's a terrible compulsion and I wish I could stop.

[@nicoleng4181]

16:36 yes i feel so accomplished hahah and omaigosh i've been finding a video that explains this well and you're the only one that I UNDERSTOOD SO WELL thank you

[@ayodelerighteous]

You just salvaged my fast approaching mathematics examination!

[@joshuastier231]

Dude you are fantastic thank you so much

[@blackpenredpen]

Joshua Stier My pleasure to help!!!

[@zhou4168]

it is a good video, thanks. Already subscribed, hoping to watch more!

[@blackpenredpen]

zhou coooool thanks!!!

[@xOxAdnanxOx]

all the way from the u world to differential equations, we been together my friend... thank you

[@photomath4327]

Exactly what have been searching for ...thank you so much 🙏🙏, your video have been helping

[@Engineering_conceptsUOM]

Thank you sir… you are my mathematics hero

[@GisselMejiashortyeah]

You are awesome. Thank you very much

[@davidnigelsemuyaba1131]

I do appreciate you for your service, you have rilly talented in teaching, may God 🙏 bless you

[@timotejfasiang]

The hardest part for this guy was distributing the - into the (A + B) at 7:35😂

[@LL-kn1de]

Wow! This is so helpful. I wish I found this video earlier! Thank you !!!

[@jamalmanking3928]

It so amazing please more videos please for the subject Deferential equation please.

[@blackpenredpen]

I have many playlists on that already. Check out my channel home page.

[@taekwondotime]

@1:30 How did you know it was going to be: *yp = Ae^tsin(t) + Be^tcos(t)* ? A bit more explanation is needed there. What happens if the equation is y'' - 3y' + 2y = 3t ? What happens if the equation is y'' - 3y' + 2y = tan(t) ? How does that piece change? Thanks.

[@gokayfem]

you should check his differential equations playlist. there is variation of parameters method for tant and other types of undetermined coefficients methods for 3t, e^t kind of things.

[@alvelymondragon33]

I think it's from a table given.. that what i use actually

[@carultch]

If the function on the right hand side is linearly independent of the homogeneous solution, you directly match it with an arbitrary coefficient, and accompany it with other coefficients on its counterparts (more on that later). If it is a constant multiple of one of your solutions to the homogeneous solution, then you multiply by t until it isn't. For exponentials, the particular solution will also be exponential. For sine or cosine, or a linear combination of both of the same frequency, the particular solution will also be a linear combination of both of the same frequency. Likewise for exponentials, enveloping the sine and cosine. For constants, the particular solution will be a placeholder constant, not necessarily equal to your given constant For polynomials of t, the particular solution will be a polynomial of the same degree, and placeholder coefficients along the way. For anything else that isn't a linear combination of the above, you'd have to use another method, like variation of parameters.

[@bukkiofuzi1725]

I love how you handled the second derivative

[@shoberino3898]

14:49 “I didn’t even use a calculator so you guys can do this too” says the human calculator

[@anamuslamh09]

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[@tomriddle9489]

great😀

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[@__IzanNafisRahman]

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[@TheGuroLOLITA]

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[@elisamarconell9991]

thank you kind sir

[@aljoker3053]

A coil of inductance L Henry and a capacitor of C Farad are connected in series if I=I0, Q=Q0 when t=0 Find: Q and I when t

[@carultch]

The corresponding diffEQ is: L*Q"(t) + Q(t)/C = 0 Characteristic equation: L*r^2 + r/C = 0 Solution for r: r = +/-i/sqrt(L*C) The solution for Q(t), is a linear combination of sine and cosine, with a frequency equal to the imaginary part of r: Q(t) = A*cos(sqrt(1/(L*C))*t) + B*cos(sqrt(1/(L*C))*t) At t=0, Q(0) = Q0, and I(0) = I0, which is the same thing as Q'(0). Find Q'(t), which is also I(t): Q'(t) = sqrt(1/(L*C))*(-A*sin(sqrt(1/(L*C))*t) + B*sin(sqrt(1/(L*C))*t)) At t=0, sine is zero, so the cosine terms govern. This means: A = Q0 B*sqrt(1/(L*C)) = I0, solve for B: B = I0*sqrt(L*C) Thus our result is: Q(t) = Q0*cos(sqrt(1/(L*C))*t) + I0*sqrt(L*C)*sin(sqrt(1/(L*C))*t) I(t) = sqrt(1/(L*C))*(-Q0*sin(sqrt(1/(L*C))*t) + I0*sqrt(L*C)*sin(sqrt(1/(L*C))*t))

[@XLatMaths]

Why don't you have to set the particular integral to A*te^t*sin(t)..., the auxiliary equation already has e^t as one of its components. Don't you have to multiply the P.I. by one power of the variable when one of the auxiliary equation's roots is in the RHS?

[@amzayd]

I have the same question!!!

[@zingaphiranana8425]

@@amzayd he made a mistake

[@ernestschoenmakers8181]

@@zingaphiranana8425 Yes he did cause at 12:25 he plugged in -A + B but it should be -A + 3B, so A and B are miscalculated.

[@oluchukwuokoye8412]

Thank you for this video

[@bandarhindi8450]

your techniques are very helpful

[@mumtazbegum3492]

Very well ....super bro... thank u so much

[@_ssodaaa]

In particular sol. Can I consider yp= Ae^t sin(t) not yp= Ae^t sin(t) + Be^t cos(t) ? Because A&B are abitary const.

[@carultch]

Only if you are "lucky" enough that B=0. For this method, you have to account for the possibility that both sine and cosine will play a role in the result, until you rule out the possibility.

[@antonellaa7120]

I UNDERSTOOD EVERYTHING!!! GOD BLESS YOUUU

[@blackpenredpen]

Thanks

[@Eta_Carinae__]

B=-1/2 not 1/2. Otherwise -A-B=0 isn't satisfied. Final answer should read: y=C1e^t+C2-(1/2)(e^t)sin(t)-(1/2)(e^t)cos(t)

[@jonhsmith3745]

-(-1/2)-(1/2) = 0

[@joshuammaina4911]

Can you please do one with the differential operator Pretty please with chocolate frosting on the top

[@aprotutor]

Operator Method: Auxiliary equation: r^2-3r+2=0 (r-1)(r-2)=0 r=1,2 Complementary solution y_0=C_1 e^x+C_2 e^2x (D-1)(D-2)y=e^x sin⁡x Particular solution y_p=1/(D-1)(D-2) (e^x sin⁡x) =e^x 1/(D+1-1)(D+1-2) (sin⁡x) =e^x 1/D(D-1) (sin⁡x) =e^x 1/((D^2-D) ) (sin⁡x) =e^x 1/((-1-D) ) (sin⁡x) =〖-e〗^x ((D-1))/((D^2-1) ) (sin⁡x) =〖-e〗^x ((D-1))/((-1-1) ) (sin⁡x) =〖-e〗^x ((D-1))/((-2) ) (sin⁡x) =e^x/2 (cos x-sin⁡x) General solution y=y_0+y_p=C_1 e^x+C_2 e^2x+e^x/2 (cos x-sin⁡x)

[@edgarb.6187]

Ive been looking for examples of X^n*sin(x) Or X^n*cos(x) But haven't seen any on youtube. Any one know some links? Thanks.

[@ITSYUNGIAN]

Thanks

[@atone0909]

Very nice

[@tyronekim3506]

Excellent video. Thanks!

[@tejagandreti]

Tq so much bro very gud explanation

[@feelingepic6087]

Thank u so so much for helping me...now i understand it clearly

[@cyberwarrior5217]

oh my god..you make it easy for me..and you r soo funny

[@jonhsmith3745]

banging video

[@kazshoichi9369]

I feel so accomplished indeed my man

[@arvandvedadi2457]

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[@bandarhindi8450]

Thanks a lot. The video helped me a lot you are awesome

[@Ayeventy]

A million times better than my professor

[@tungamiraiaaron1784]

u are the best

[@cadendennis3019]

That was awesome

[@icee562]

wow, EXTREMLY helpful!!!!!

[@lorenmanchester3377]

Thanks so much! This video really helped me understand.

[@hapias9300]

Thank you well explained.

[@olivierbkofficial9426]

My God!!!! you're just amazing ,thx

[@elnathanlynd5418]

Amazing

[@tomatrix7525]

Unbelievable!

[@himanshubanate9061]

nice sir thank u so much

[@wraster07hp36]

Thanks bro.

[@eddiekanhai]

Thank you so much, this helped me a great amount !

[@Dr.Reactor]

channel name: blackpenredpen Blue pen: Am I a joke to you??

[@arnelmananghaya7588]

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[@deorum59]

Not all heroes wear capes... Thanks

[@robbiemayer3573]

the literal goat

[@wzablonkalivo2818]

great work. no need to attend to attend to my lecture

[@debbiekalima4541]

thank u so much

[@jacksoncarter6352]

Perfect thanks

[@ahmedbelloe3923]

I used De-operator method and it was much easier and fast too. I had the same answer

[@josemuygay8851]

thanks for uploading this!

[@terrencemadanhi8833]

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[@hayatomcreverse7925]

Bai maayo kayka this helps a lot T.T

[@bhaswatighosh3969]

Can u solve some hardest problems in nth order eqn

[@jacksantoro182]

thank you very helpful

[@bakkamanthulamohansai3430]

Super super super

[@F19991]

Another useful video. But I think it's quite dangerous that you're holding a thermal detonator haha

[@Goosebump_guy]

How could differential Ae^t(sint) still be Ae^t(cost)+Ae^t(sint)?????dont u have to differentiate e^t into te^t ???

[@AndromedaIX]

You mentioned it was the superposition principle. Is it the same method as used for circuit analysis?

[@carultch]

It's a similar concept. Essentially, to simplify the equation, you break it up into two sub-equations that both also need to be satisfied, and work out how to satisfy both parts individually. And then form a solution that is a linear combination of the two.

[@farissh3924]

thanku

[@mtahir9181]

Thank You.

[@Channel-uf6ko]

I can't find any video on the internet doing a combination of polynomial, exponential, and sin and cosine :((

[@pushpanjalinayak2765]

Thank you so much

[@Cannongabang]

hello blackpenredpen... is the integral sqrt[(x^4-1)/(c-x)] dx solvable???

[@DaQwertyKidNL]

No

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blackpenredpen used blue pen!

[@rymichael7203]

I'm a year to late.

[@lesegomokgabudi5798]

thanks bro

[@strongerwithstubsstretch3472]

thank you sir!

[@user-yw8pr8wp4b]

Yp= t e^t ( A Cos(t)+B sin(t)) IS true ?! Because e^t IS in Yh and in f(t) = e^t sin(t) ?! Please answer me

[@AstroRoxy]

Thanks dude!

[@dmorgan0628]

I'm taking calc2 again this summer and diffeq this fall. This video is scary!

[@cyberwarrior5217]

Dan Morgan hahahha summer..I hate summer

[@mastermfundisombhokane5731]

maVuti maths 3 where are you? send a shout out!

[@thomasjefferson6225]

I sincerely hate these problems

[@TheIvar19]

I just don´t get why we need the Yh solution. Isn´t Yp enough? What does the Homogenous solution add?

[@carultch]

yp only solves a special case of initial conditions, that the coefficients in front of the yh parts diminish to zero. The yh part is the transient response from initial conditions, while the yp part is the steady state response that governs what it will be, once the initial conditions are effectively out of the picture. As an example, given: y" + 4*y' + 3*y = 10*cos(t) The homogeneous solution is: yh = A*e^(-t) + B*e^(-3*t) The particular solution is: yp = 2*sin(t) + cos(t) In the special case that our conditions start on the curve of yp, then the coefficients A and B both become zero, and we don't need the homogeneous part. This would happen if, in this example, y(0) = 1, and y'(0) = 2. What the homogeneous part adds, is the ability for the system to start at any set of initial conditions, and exponentially decay to eventually match the particular solution.

[@wajeehalamoudi5945]

is there any previous illustration video such #18 or 4?

[@sunitalymon1925]

him: whenever there's sin we must have cosine me: say no more.

[@FS-zt6tm]

what would the general formula be for yp if the right side is e^x(6cosx +17sinx). I'm struggling with trig functions. I know how to do the derivatives, I'm just not sure what to start with.

[@carultch]

Given an RHS of e^x * (6*cos(x) + 17*sin(x)) The general form of the yp particular solution would be: e^x * (A*cos(x) + B*sin(x)) As long as your given linear combination of sine and cosine both have the same frequency, they both correspond to the particular solution being a linear combination of the two. The e^x applies to both of them. Also important is that we don't overlap the homogeneous solutions, otherwise we'd need to multiply by t until it is linearly independent. As an example: y" + y = e^x * (6*cos(x) + 17*sin(x)) While this may look like there is overlap, there really isn't. We'd have to have y" - 2*y' + 2*y = e^x * (6*cos(x) + 17*sin(x)), to have overlap, since the homogeneous solutions of this one, also include the exponential envelope on the trig functions. The homogeneous solution: yh = A*cos(x) + B*sin(x) The particular solution: yp = e^(x) * (C*cos(x) + D*sin(x)) Differentiate twice: yp' = e^x * ((D - C)*sin(x) + (C + D)*cos(x)) yp" = 2*e^x * (D*cos(x) - C*sin(x)) Apply to original diffEQ, and equate coefs to find C & D: 2*e^x * (D*cos(x) - C*sin(x)) + e^(x) * (C*cos(x) + D*sin(x)) = e^x * (6*cos(x) + 17*sin(x)) Cancel common e^x factor: 2* (D*cos(x) - C*sin(x)) + (C*cos(x) + D*sin(x)) = 6*cos(x) + 17*sin(x) Thus, equating coefs we get: 2*D + C = 6 -2*C + D = 17 Solve for C & D: C = -28/5, D = 29/5. Thus the solution is: y = A*cos(x) + B*sin(x) + e^x*(-28/5*cos(x) + 29/5*sin(x))

[@liujinpeng2635]

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[@ghozmit1]

Bro, you picked the hardest/longest example :(

[@blackpenredpen]

ghozmit1 i have done other simpler problems too

[@ghozmit1]

oh I see, Thanks :)

[@icee562]

trust me, one you see a hard(er) example, the others are cake!

[@jammcrusader1981]

if you can swim in the deep end you can swim in the shallow end my friend

[@Sneaky1ne]

the only reason I clicked was because it was a harder problem

[@AdnanAdnan-hd7co]

ارجو حل الكثير من المعادلات التفاضليه

[@zabihullahhalim5096]

that is great :) T H A N K YOU

[@sabikasarwat4186]

What should be the method of solving this equation D2y/dt2 _ dy/dt _2y= 5 cosh(2t)....by undetermined method....kindly help plzzz

[@carultch]

Assuming your underscores mean +, we're given: y"(t) + y'(t) + 2*y = 5*cosh(2*t) You can use undetermined coefficients, since cosh is just a linear combination of two exponentials. I'll assume a particular ansatz, of a linear combination of cosh and sinh. First find the homogeneous solution: yh" + yh' + 2*yh = 0 (r^2 + r + 2)*e^(r*t) = 0 (r^2 + r + 2) = 0 r = (-1 +/- sqrt(1 - 8))/2 r = -1/2 +/- i*sqrt(7)/2 Thus: yh = e^(-t/2) * (A*cos(t*sqrt(7)/2)) + B*sin(t*sqrt(7)/2))) For the particular solution: yp = C*cosh(2*t) + D*sin(2*t) Apply to original diffEQ, and match coefs: yp' = 2*C*sinh(2*t) + 2*D*cosh(2*t) yp" = 4*C*cosh(2*t) + 4*D*sinh(2*t) Thus: 4*C*cosh(2*t) + 4*D*sinh(2*t) + 2*C*sinh(2*t) + 2*D*cosh(2*t) + 2*C*cosh(2*t) + 2*D*sinh(2*t) = 5*cosh(2*t) 4*C + 2*D + 2*C = 5 4*D + 2*C + 2*D = 0 solution for C&D: C = 15/16, D = -5/16 Thus, the solution is: y = e^(-t/2) * (A*cos(t*sqrt(7)/2)) + B*sin(t*sqrt(7)/2))) + 15/16*cosh(2*t) - 5/16*sinh(2*t)