Variation of Parameters for Differential Equations (KristaKingMath)

[@aliozanerbektas]

I took the same class for 6 semesters, yet what my "professors" could not teach me in 6 semesters, you taught me in 6 minutes... I wish I had a way to make your life easier as you did mine! I love you!

[@kristakingmath]

I'm so glad I could help, Ozan! :D

[@zoom2121]

OMG yes you made a video on variation of parameters!!! My request didn't go unheard lol Seriously, we all really appreciate the time and effort you take to make these awesome videos! Thank you :)

[@theonevic1]

The way you dummy down this things is amazing.what can i say more.thanks alot from the bottom of my heart.i pray sooner than later someone will see what you have been doing and really bless you.

[@GamerwithaConscience]

You've marvellously explained this time consuming topic in such a lucid way unlike, other KZbin video tutorials which make this topic cumbersome

[@kristakingmath]

+prateek sriv Thank you very much, I'm glad you liked it!

[@ahexcuseme6936]

You have saved my life so many times, Ms.Krista. Many thanks!

[@rafaelgpontes]

You're seriously a life saver! The world needs more people like you! Keep up the great work, girl. Cheers from Brazil.

[@kristakingmath]

congratulations!! way to go! i'm so happy for you!! :D

[@SPECHALAIGENT]

Thanks for the amazing explanation. You're my Math superhero.

[@k4kfh]

Thank you so much! 50 minutes of lecture and I was lost. 10 minutes into your video and I'm already understanding better. Krista for president

[@kristakingmath]

You're welcome!! Thank you for watching! :D

[@jflopezfernandez]

Krista, you have no idea how hard this video saved me. Awesome video, thank you so much, you really are the best

[@kristakingmath]

Aw thanks! Glad it could help. :)

[@Uncertaintycat]

Best part about Krista's videos: She can explain in 4 videos what my book takes 4 chapters to get across. That's less than 1/2 an hour versus 2 weeks! This means when I go to take the test, it is all chopped up in the 4 parts because they assume it was really hard and I probably didn't get it. However, since I watched these, the test feels like it's all extra credit!

[@dogeness]

The books take longer to explain it because they actually prove all the formulas and theorems. Krista just gives you the bare minimum to be able to plug and chug the problems out. Not that that's a bad thing. This saved my ass an hour before my exam lol.

[@marthabarranco9423]

God bless your soul for these AMAZING videos!! They're so clear straight-forward! Thank you for using your gift in such a kind way and sharing it with the rest of the world!:)

[@jayadams35]

Thank you Krista you just saved my 3 hours lecture in just 13 precious minutes. I love you so much.

[@kristakingmath]

You're welcome, Jay! Happy to help! :)

[@mathlover2299]

Just want to say thanks. I watched your videos all through my time at community college. I am transferring this fall to UC Berkeley for Physics. Could not have done this without your help and other KZbin tutors. Thank you very much.

[@NolanKray]

Informative, clear, and direct. Well done!

[@cesaraguilar4016]

Really helpful video!! Made this sound easy for the most part, my professor thought it the same exact way, but your clarity was much more helpful! Again thank you for people like you that understand every detail is important to clarify!

[@kristakingmath]

+Cesar Aguilar You're welcome, I'm so glad I could help!

[@Zakaria31121986]

Krista, you are the KING! Thank you for the clarification. My math teacher is a great teacher but your explanation is way more clear!

[@kristakingmath]

I'm so glad it helped! :)

[@owenkelly3043]

the only variation of parameters video that is explained in plain english! thank you

[@RizA-t2n]

thanx a zillion..I had my lecture & cudn't understand a word..but u made it look so easy & plain..

[@kristakingmath]

+Muhammad Rizwan Aziz I'm glad I could help!

[@kristakingmath]

I'm just happy I can help in my own little way. :)

[@kelvinfeliciano]

Hey. I just want to say thanks for taking the time to help others. We are doing it in a different way. But, I think it will be a lot easier with the formula you are using.

[@kristakingmath]

Awesome! I'm so glad I could help! :)

[@cypress1173]

Differential Equations is such an interesting class. I'm glad I took it

[@ThilebanTheEngineer]

It's fun to watch :D

[@kristakingmath]

I'm so glad it helped!! :D

[@salmanqureshi3361]

I was literally stuck on this topic for this entire year and then I came across this video. Thanks for saving my life and great video I'll definitely be sure to check out your other videos ❤️🙏🏽take care

[@kristakingmath]

I'm 26. I've never been a professor, just have tutored on the subject for a long time. :)

[@kristakingmath]

you're welcome!! i'm so glad it helped!!! :D

[@abdul9525]

Miss, I don't know your name, but you are amazing. You have helped me a lot in my calc 3 class and you are helping me now with diff equation. You made me save a lot of money and effort, and I just want to let you know that your work is highly appreciated. Thank you, thank you very much from the bottom of my heart.

[@kristakingmath]

You're very kind. Thank you so much for letting me know that the videos have been helpful!!

[@anaskalkal1219]

only 4 hours left to the exam, I am lost without you and Patrick videos thanks a lot

[@kristakingmath]

Hope you rocked the exam!

[@anaskalkal1219]

Similarly to that example, there was one and I've solved easily :) Thanks Krista

[@PLP0321]

Awesome! Followed step by step and never got lost for a moment. Thank you!

[@kristakingmath]

Yay! That makes me happy. :)

[@kristakingmath]

yes, and your complementary solution would be (c_1)e^(-x) + (c_2)e^(x) + (c_3)e^(6x), and you'd take it from there. hope that helps! :)

[@JB-kb6jw]

wow! the whole chapter summarized in 11 min, ! thanks !!!

[@Jerry-vd9zm]

my textbook and notes gave few examples of them , thank you a lot for explaining the variations of parameters, much appreciated, love you!

[@booysenbafana4358]

Awesome... My lecturer hasn't taught me this. All thanks to you I can do all my Tutorial problems

[@theonevic1]

How does it feel to know that you are actually a blessing to your generation?you are truely a plus. i have amazed my friends with this fomula you provide us,you always find a way to make things soooooo easy.Bill gate will soon discover you.that way you have a better plat form to reach the world .blessings Madam,

[@faisalal-balawi1473]

Thank you very much, that was really helpful . every time I have an exam I search for your videos , and you really save me thanks

[@kristakingmath]

Thanks for letting me know! I'm so glad I've been able to help. :)

[@stephanm.tjaden3887]

Yeah, Krista, You are awesome. You make complex problems so much easier to understand. Thank you!

[@kristakingmath]

You're welcome, Stephan! I'm so glad the videos are helping! :D

[@personwithskills]

Your explanation is so clear, and you predicted all my questions about this method. Thanks so much!

[@kristakingmath]

that's awesome! i'm so glad it helped!

[@uhuihiuhuihi]

Have an assignment due in a couple hours, was using another method from another video that wasn't getting the right answers. This video has just saved my mark. Thanks for the help :)

[@kristakingmath]

You're welcome, I'm so glad it helped! :D

[@rjknutsen]

when u get two complimentary solutions y_1 and y_2 that are the same (as in this case; we get e^x twice), you want both solutions to be linearly independent, so that when you take the wronskian later, they won't equal zero (among other reasons). So we multiply e^x by a factor of x. For three repeated roots you would get x^2e^x, and so on.

[@FaizanAhmad-vl1tp]

Thank a lot of you for providing an easy method to solve variation parameters.

[@jonathancolina7772]

YOU ARE AWESOME!!! You explain a whole lot better than my PhD teacher. ; )

[@kristakingmath]

You're welcome! I hope it helps!! :D

[@kristakingmath]

Eesh... that's a big detail to skip! I'm so glad this helped!! :D

[@kristakingmath]

You need the x in the second term in order to distinguish it from the first term. You have to include that x whenever you have equal real roots. Hope that helps clarify! :)

[@oscarmrtinz8879]

Thanks... ❤️

[@mooshicat]

I don't know how my instructor went through this whole section without explaining how she found the denominator (didn't hear the word Wronskian once!). This makes a whole lot more sense now, thanks!

[@Drew-gi5yb]

i am already in love with the way you teach ,super fantabulous !!

[@kristakingmath]

Thank you so much!

[@MrDTraff]

You're fantastic. Probably saved my Calc 4 career. :)

[@kristakingmath]

yes, if the roots are not equal, then you use r_2 in the second term. also, you wouldn't need that x anymore, because you can distinguish the terms with r_1 and r_2, so the x becomes unnecessary. :)

[@kelvinfeliciano]

You are not going to believe this but, I have only subscribe to you. You deserve it. keep up the good work.

[@epharathapalo9617]

Hi lady you making me so feel good when I see your explaining. I am fully understood!

[@kristakingmath]

I'm so glad! You're welcome!! :)

[@gokaygg5809]

Wow that was great haha! Better than my professor and 2 calculus books. Many thanks!!

[@kristakingmath]

Hope the midterm went well!! Glad I could help. :)

[@prabhjitmultani1596]

omg thank you so much, u just saved me on my exam today. I'm doing so well in my class because of your videos. Thank you.

[@kristakingmath]

That's so awesome! Thanks for letting me know. :)

[@kristakingmath]

Undetermined coefficients here: integralcalc(dot)com/second-order-non-homogeneous-differential-equations/. Hope that helps! I'll try to keep adding more videos! :)

[@sonuuk1]

you are the best! can't thank you enough for everything that you are doing for us. love you! God bless you.

[@galen21]

Undetermined Coefficients works great when your g(x) [whatever the DE is set equal to] is exponential or cosx etc..... however, when you get things like lnx, tanx, cscx you will run into problems. Variation of paramenters will always work and can even be applied to higher order DEs - it's just messier so undetermined coefficients is good to use when possible so there is less room to make small algebraic mistakes.

[@brighttsikira6310]

This is so clear and straight forward .Big thank you

[@kristakingmath]

You're welcome, Bright! I'm so glad it made sense! :)

[@kevinlafuente9087]

hi Krista! i just used, to get particular solution to the same problem above, these methods: 1. solving y(x) by C1 and C2 with U1 and U2 respectively. It will turn out like this y(x)= u(1) + u(2)e^(-2x). Then determine the unknown u's by getting the 1st and 2nd derivative of the new y(x), then solve them simultaneously. (I have watch it using to solve 2nd orders from different video) 2. solving yp(x)= -y(1)∫[y(2)*g(x)]/W(x,y) + y(2)∫[y(1)*g(x)]/W(x,y) (seen from your seperate vid Variation of Parameters for DE) using different method I got the same particular solution: 6x^2 -6x -xe^(-2x)/2 -e^(-2x)/4 +3 I got the last two more terms which are not in your solution. So, is this particular solution also valid? or it is definitely wrong? and if it is, have I used the methods improperly? or the methods I mentioned are just not valid for the problem concern in the video? I really like your videos. They're easy to understand and are reasons why I enjoyed solving DE now. Thank you for all the enlightenment. :)

[@khushlamba9257]

Thank you so much for explaining this method. I have a test in a few days that I'm studying for and I really needed to learn this technique.

[@kristakingmath]

You're welcome! Good luck on your test!!

[@kristakingmath]

I'm so glad I could help! :)

[@emrefatihCakmakci]

you are very very good teacher..When my maths teacher explain in classroom.I did not understood.but I watched your video.and I understood very well.Thank you so much..please contiune help to us

[@kristakingmath]

Yay 15 points! :) Sorry about your textbook, but I'm glad I was able to help.

[@kristakingmath]

Awww!! You're welcome! You made me smile!! :D

[@davidson2727what]

wish I saw this last night before my quiz this morning. Subscribed now.

[@kristakingmath]

Next time at least! Thanks for subscribing. :)

[@metazoxan2]

So would I have to do a 3 column wronskian? Also for the second step with the integration how should that be set up? First there was F1 with an F2 inside the integration then there was an F2 with an F1 inside the integration. But if there is a thid one how does that get set up?

[@A58i2]

BEST explanation i ever came across, THANKS Maam

[@kristakingmath]

:D

[@ctarhan]

Thank you Krista, clear and very comprehensible video. :)

[@DBTING]

Hi, say the comp solution is e^-1,e^2,e^3... do we have to arrange it from smallest to the largest in the Wmatrix? because if we simply put it, then the answer would be different.

[@firstnamelastname9846]

you know ur really good when you have 0 dislikes on youtube. you explained this Very well! thank you

[@kristakingmath]

If only I could say that for all of my videos! :D

[@firstnamelastname9846]

integralCALC i honestly dont know how any of ur videos can get any dislikes. ive seen many of your videos and i could understant everything very well.

[@BusyNames]

Well, this was waaay easier than Wikipedia or any other readable material. Thanks a lot!

[@andrewgarcia3967]

Does the coefficient have to be constant for the Variation of Parameters method?

[@Ravenex2]

Very good video, taking Differential Equations at Rutgers university. This made the section really digestible for me.

[@Robtmmartine77]

Is there a form of this formula for third and higher degrees?

[@leonthethird7494]

Yes

[@Tr0IlFace]

hi, I have a question... Does it matter which value I use for y1 or y2? because y1 and y2 crossing each other would get different result?

[@Maldioso]

This tutorial helped me pass my ODE thank you so much ❤

[@kristakingmath]

you're welcome, i'm glad it helped! :)

[@feysalimran]

Is it right for me to assume that we use this method, "Variation of parameters" when we got a rational functionon the right hand side of any non-homo D.E?

[@bigboibill7854]

Couldn't ask for a better video - thanks!

[@kristakingmath]

😊

[@SoloPro]

best videos ever

[@kristakingmath]

You are so welcome!

[@jiwonkim9001]

Hi Krista, I am wondering how I can solve this if I have a higher order diff equation?

[@amjad-se]

tutorial.math.lamar.edu/Classes/DE/IntroHigherOrder.aspx This might help you

[@kristakingmath]

Glad you think so! :)

[@HanitpalSingh]

imo this lesson is much easier to understand/remember if everything is taught in matrix form. the wronskian matrix you have, but even the integrals can be set up as iteg([0 y2,f/a y2'] ) + integ([y1 0,y1' f/a])

[@JordanMedina]

Hello, I have a question. At around 9:00 you didn't cancel out the x in front of your integral with the integrand which is (1/x^4). Why isn't this proper? Thanks so much =D.

[@ar15is223]

Great video Video...It definitely got me 15 more points on a test than i would have got if i had not seen it. Also I think you do a great job of explaining how to do the problem in a simple way than in the way my textbook explains is.. My textbook is Elementary Differential Equations and Boundary Value Problems and i dislike it because it skips to many steps in example problems and the student solutions manual is junk.. Thank you foir putting the effort it takes to create these videos

[@metazoxan2]

You sure beat the hell out of my text book. This will be an easy section thanks to you.

[@kristakingmath]

You saved it by putting in the time! :)

[@Ensign_Cthulhu]

Outstanding. In particular, you make the process of calculating the Wronskian and the particular-solution terms seem a lot less scary than it first appears. The key is to remember Yc(x) = c1y1 + c2y2, where g(x) comes from, & the exact form of the Wronskian determinant, and it all goes smoothly after that (provided your basic algebra and integration skills are up to scratch, of course!). I assume when the roots of the complementary equation are complex that y1 = e^ax cos bx and y2 = e^ax sin bx?

[@Mohammad62881]

u r at least 10 times better than my professor thank you :)

[@rawBreh45]

You're amazing! Between this video and your vids on undetermined coefficients I've learned what took my professor weeks to explain in class. #instantsubscribed

[@kristakingmath]

+Ryan Najmi Awesome, thank you so much Ryan!

[@MariAmmaSar]

What are the limits of this integration, and how do these limits differ when this method of variation of parameters is used in the full Fourier transform or the Fourier sine or Fourier cosine transform?

[@carultch]

There are no limits of the integrals in variation of parameters. They are indefinite integrals. If you're wondering what happens to the arbitrary constants of integration, they ultimately gets absorbed in the two arbitrary constants on the homogeneous part of the solution. I'll give an example to show how this works. Consider y" + y = sec(x) Homogeneous solution: yh = A*cos(x) + B*sin(x) Particular solution: yc = -cos(x)*integral(sin(x)*sec(x)) dx + sin(x)*integral(cos(x)*sec(x)) dx The integrals evaluate as: integral(sin(x)*sec(x)) dx = integral tan(x) dx = -ln(|cos(x)|) + C integral(cos(x)*sec(x)) dx = integral 1 dx = x + D Composing the particular solution: yc = -cos(x)*(-ln(|cos(x)|) + C) + sin(x)*(x + D) Forming and factoring the general solution: y = (A - C)*cos(x) + cos(x)*ln(cos(x)) + (B + D)*sin(x) + x*sin(x) As you can see, no matter what we opt to use for C and D, they will just be absorbed in the constants A and B, when we use initial conditions to solve for them.

[@muhammadnawawi212]

This video is really awesome..help me so much solving my homework..thanks you for your good job

[@Knightmetal]

I have a little confusion, if instead of two C's in my homogeneous solution (C1 and C2) I have three, e.g C1 + C2*exp(2x) + C3*exp(-2x), what will my Wronskian matrix look like? will it be a 2X3 matrix? how do I solve it if that's the case. Nice videos by the way :)

[@AKDMB]

What is considered V1 and V2 in this approach? I can solve using variation of parameters, but some questions ask to find V1 or V2, what term or terms are considered V1 and V2?.

[@gokaygg5809]

I have a question; if there was a third order differential equation to begin with, for the particular solution would the third integral be a plus or a minus? You did -Y1(integral)+ Y2(integral)...... what would be the signs of the third and fourth integral? Does it always go -,+,-,+?

[@carultch]

If there were a 3rd order diffEQ like this, then you'd have three variations on the Wronskian, which would be W1, W2 and W3. You'd also have a master Wronskain in the denominator, that is a 3x3 determinant. Row 1 = original homogeneous solution components, Row 2 = their first derivatives, Row 3 = their 2nd derivatives. You'd then swap out the corresponding column to make W1, W2, and W3, for a column consisting of mostly zeros, and a 1 as the final entry. I call these the Cramer Wronskains, since they ultimately come from Cramer's rule.

[@carultch]

As an example, suppose we had: y'" + 6*y" + 11*y' + 6*y = g(t) The characteristic cubic of (r^3 + 6*r^2 + 11*r + 6) factors as (r + 1)*(r + 2)*(r + 3), so the homogeneous solution is: yh = A*e^(-t) + B*e^(-2*t) + C*e^(-3*t) The Wronskian W = | e^(-t) _____ e^(-2*t) ____ e^(-3*t) | | -e^(-t) __ -2*e^(-2*t) __ -3*e^(-3*t) | | e^(-t) ___ 4*e^(-2*t) ___ 9*e^(-3*t) | W = -2*e^(-6*t) The particular solution will be: yp = e^(-t) * integral W1*g(t)/W dt + e^(-2*t) * integral W2*g(t)/W dt + e^(-3*t) * integral W3*g(t)/W dt W1 would be: | 0 _____ e^(-2*t) ____ e^(-3*t) | | 0 __ -2*e^(-2*t) __ -3*e^(-3*t) | | 1 ___ 4*e^(-2*t) ___ 9*e^(-3*t) | W1 = -e^(-5*t) And the other two Cramer Wronskains would be found in a similar way, swapping the corresponding column for a column of 0, 0, 1.

[@jenkinsj9224]

will i get a different answer if i switch the order in which the fundamental solution set is written?

[@sangamkhadka6221]

Really got it at first time 🔥best teacher

[@kristakingmath]

Thank you so much, Sangam, I'm so glad it made sense! :)