Second-Order Non-Homogeneous Differential (KristaKingMath)

► My Differential Equations course: www.kristaking...
Second-Order Non-Homogeneous Differential Equations calculus problem example.
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If you could use some extra help with your math class, then check out Krista’s website // www.kristakingm...
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Since then, I’ve recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math student-from basic middle school classes to advanced college calculus-figure out what’s going on, understand the important concepts, and pass their classes, once and for all. Interested in getting help? Learn more here: www.kristakingm...
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[@kristakingmath]

I always hated when the details were skipped over, so I do my best not to. :)

[@muayyedsiraj]

Thanks thats true 👍🏽

[@NISHANTKUMAR-l6x]

thank you for your video 😃

[@sanlistonmadzima]

Thank you so much. I'm a particular fan of how you show every single step, even the arithmetic. I really appreciate that. I am subscribing.

[@kristakingmath]

+waytoomanyllamas Thank you, I'm so glad you like the videos!

[@TheDas1888]

Yep, many teachers just brush over the steps too quickly but this video really broke down the steps and made it much easier to understand at a deeper level. Thanks!

[@kristakingmath]

You're welcome! :D

[@djm6683]

these vids from almost a decade ago are still the absolute best

[@kristakingmath]

@zoom2121 You're so welcome, and thanks for the compliment! :D I'll continue adding more DiffEQ stuff as quickly as I can! :)

[@brandoncastaneda123]

This channel has been the biggest help in all of my college math courses for the past couple years. You're the best!

[@kristakingmath]

+Brandon Castaneda Aw thanks! I'm so glad I've been able to help along the way!

[@bronwynlaird6053]

you even reply to everybody's questions. wow you are truely amazing.

[@kristakingmath]

Maybe not every single one, but I try to get back to as many people as I can!

[@kristakingmath]

@SirArghPirate I will try to get to boundary value problems when I can, but I have to be honest, I'm trying to focus on getting all of the videos up for calculus 1 and 2 before I move on to differential equations. Either way though, I'll post a BVP as soon as I can! :)

omg thank you so much for this, please continue making videos, you have no idea how much people you are helping through this, I even told some of my classmates about your channel, keep up the great work!!

[@kristakingmath]

thank you so much! i'm so glad the videos have been helpful, i hope you'll continue to use them!

[@kristakingmath]

If you subtract that portion, then it just moves from the right hand side to the left hand side, and you're still looking at a nonhomogeneous DE. Homogenous means you've got only y's on the left, that are set equal to 0 on the right.

[@kristakingmath]

Thank you so much, Michael! I really hope your students will find my videos helpful when it comes to mastering differential equations! :D

[@kristakingmath]

In this video, I'm using sketchbook to write, and screenflow to record my screen. :)

[@lukasodern]

Finally someone who explains why you sometimes have to multiply the particular solution with x! Thank you!

[@linkhidalgogato]

someone is here for finals

[@lukasodern]

@@linkhidalgogato You betcha!

[@kristakingmath]

Hope the final went great!! :D

[@edukate_eng6541]

So so so helpful! Very well explained. Especially the bit about the particular solution and multiplying by x to get rid of overlapping. Thank you!

[@kristakingmath]

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

[@JimmySmithJunior]

Awesome video. Learned more, and learned better, in this video, than a 50minute lecture from an accredited mathematician. Thank you.

[@jandon603]

I've watched your videos since calc 1 and you still help me to this day! Diff eq can be tricky but you make it so easy to understand!! Thank you for your contributions, the older students like myself appreciate all your dedication!

[@kristakingmath]

That makes my day! I'm so glad that I've been able to help along the way... thank you for telling me! :D

[@ponyluvr327]

Thank you so much for all you do on this channel! These are the best examples that I have seen online and there is more than one question so we can check if we actually know it. I have told all my friends about this channel and they all love it too. Thank you for all the time that you put into this! It is because of you that I have hope on my final. Please keep posting these amazing videos!

[@ahmedmusa517]

without you and professor Leonard channel 80% of students would never pass calculus exams you are the real teachers

[@kristakingmath]

I'm happy to help, ahmed! :D

[@kristakingmath]

it would change it, but that's exactly what you want to do. remember that, at that step, you're trying to figure out what the particular solution IS, and you haven't settled on the answer yet. so you're not changing something that's already correct, you're trying to figure out what the correct solution is. multiplying by x is something you have to do to get to the correct particular solution. hope that helps clarify things... :)

[@kristakingmath]

@ouzairhafeez Great idea! Maybe I'll try that next time I do a video with a lot of equations. :) And thanks! I'm so glad you liked the video! :)

[@ErikFanki]

Just found that free e-book of yours as well - great stuff - much better than the advised textbook for my course, which I think has a goal of actually making it deliberately hard to understand. I do admire people who creates these kind of things for free - keep up the good work! :)

[@crispyhaole8533]

Wow. Just Wow. 21 minutes to cover what my professor failed to in a 3 hour lecture. More detailed, more cases covered, and much more clear. I'm better off cutting class and watching you. You are an ANGEL!

[@kristakingmath]

I'm so glad it could help! :D

[@catfran7250]

So so so helpful. I missed the day our lecturer covered this topic so I had no idea what it was about, so this is great!

[@kristakingmath]

If you have x's on the left, you want to separate them and move them to the right if you can. If you can't separate the x's from the y's you may need to use the integrating factor method to find the solution to your DE. :)

[@addy7464]

One of the best math teachers on youtube..... Thanks for these lectures.

[@HotsumaSadamoto]

This was very well done! Bravo

[@kristakingmath]

Thanks, K!

[@kristakingmath]

I DID use product rule! :) The last two terms of the second derivative are the derivative of -2Cxe^(-2x), not just the 4Cxe^(-2x). I hope that makes sense! :)

[@metazoxan2]

Thank you so much. I'm working ahead because we are having a big project soon and I want to get my homework done early so I'll have more time for the project. But working ahead means I have to teach myself using the book and that stupid thing barely explains anything right. But you made it all so clear and you used a nicely complex example to show how it would look unlike my book which would use the most general and useless example possible.

[@jdawe02]

I can’t understand the math without your videos.. Thaaaanks a lot.

[@herrzyklon]

My problem was a little different, based on a similar issue though (y"-3y'+2y=6e^2x +4x) I used your logic and although I personally didn't multiply by x (when I did I got nonsense), I got a straightforward set of values for A, B, and C. Thanks again, you ROCK!!!!!

[@airatenor]

I'm in ordinary differential equations right now and my teacher is the pits! He never explains the reasoning behind how we solve things, but I'm very interested in the math (seeing as it is my major). Thank you so much for these videos, your instruction is wonderfully thorough and helping me stay afloat through an awful professor.

[@kristakingmath]

I know how hard it is to get through a class when your professor isn't good, so I'm glad that these videos are helping to fill in some of the gaps! Hang in there! :D

[@theastronomiefreak]

Wow! Thank you so much! Ive been looking through youtube for hours not finding anything useful, until now :)

[@kristakingmath]

@Romankidful You're welcome! Glad you found it easy to follow.

[@kristakingmath]

Think about it like you're just trying to get a template together in the format that you'll need your particular solution. You're building the template for your particular solution from the ground up, and in this case, you have to multiply it by x just to have a chance that it will work. The template isn't finished until we say it's finished, and in this case, that's not until after it's multiplied by x.

[@kristakingmath]

Glad you liked the video. :) The program is AutoDesk Sketchbook

[@lfcforever]

simply WOW!! There are many out there who know their stuff but haven't got the slightest clue how to explain. You however, are surely gifted! Never could pinpoint your accent though, Canadian?

[@TheCoconutBlues]

You made more sense than my professor did in 3 lectures. Thank you so much, I wish I could give you a big hug :)

[@kristakingmath]

+Danny Kwak I'm so glad I could help!

[@itzdaman]

that means you have a homogenious solution of y(x)=c1+c2+c3*e^4*x because the first two terms are multiplied by e^0 which is 1.

[@ura239]

That was a very good explanation, I always found this type of problem disturbing. .. thank you so much.

[@kristakingmath]

You're welcome, I'm so glad you liked this!

[@ura239]

:)

[@kristakingmath]

I'm so glad you liked it!! :D

[@shubhamjain8516]

I was struggling over this type of problems where we find duplicate solution while guessing. It was very well explained by you. Things got very clear to me now. Thanks.

[@kristakingmath]

You're welcome, Shubham! I know getting rid of those duplicates can be tricky, so I'm so glad the video helped clarify it! :)

[@cigrero]

I wish u cud be my personal teacher. U made it so easy to get...the steps, explanation and all. God bless

[@S0METHINGS0METHING]

Awesome example. This cleared up some gray areas for applying the modification rule. You have a gift of relaying instructions effectively, perhaps you are an aspiring Engineering Professor/Prof. of Mathematics?... Thanks so much.

[@kevinlafuente9087]

hi Krista! i just used, to get particular solution to the same problem above, these methods: 1. solving y(x) by replacing 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 separate vid Variation of Parameters for DE) and I got the same particular solution: 6x^2 -6x -xe^(-2x)/2 -e^(-2x)/4 +3 It have 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? P.S. 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. :)

[@JadAzido]

You are legendary! I have a test in 6 hours and you simplified a whole chapter for me :D

[@kristakingmath]

Awesome! Good luck on your test!!

[@ricardozuniga8710]

You are awesome! I have a test coming up and neither my book nor my professor had really helped me fully understand what to do in the case of duplication. Looks like there's a new Khan in town ;) Thanks Krista. Subscribed.

[@kristakingmath]

Glad I could help! Thanks for subscribing. 😊

[@dillongoral4419]

Really good video, I didn't pay attention in class at all and i learned in half the time. Thank you!

[@calvinla5641]

Great video! Cramming for all the differential equations material for my final that's in 10 days and I can say this is the most helpful channel I've stumbled across.

[@kristakingmath]

I'm so glad I can help! :) Good luck on the upcoming final!

[@juliapaler4845]

You are so good at explaining things... Thank you so much!!!! Please never stop making these videos!!!

[@kristakingmath]

Thank you so much, Julia! I'm so glad you like them! :D

[@Asrlex95]

I feel like I should thank you for such a good lesson. I swear diff. equations were the bane of my life. This really helped me out!! I hope this helps me pass my Calculus final next week.

[@kristakingmath]

I hope it helps too! :)

[@onimusha542]

Man I wish I found this sooner... My midterm is in an hour. The book I'm using jumps so many steps when they solve their examples that I can't really follow what they are doing. This helped me understand it better but not sure if any last minute studying is gonna help. :(

[@MarRenshai]

I asked earlier about another program you were using, but now I would like to know what this program is also. They both seem so cool. I may be looking at being a TA at my university and I feel these programs might help. Thank you for all your help. It has been a while since I did this stuff.

[@kristakingmath]

AWESOME!! I'm so glad!! Good luck on your final, I will keep my fingers crossed for you!! :D

[@kristakingmath]

@S0METHINGS0METHING Thanks so much for the kind words. I'm glad I could help. :) While I love teaching here on youtube, I currently have no aspirations of being a professor (but you never know :).

[@kelvinndereba5423]

Thanks for the video,9 yrs down the line elevating our knowledge

[@kristakingmath]

You're welcome, Kelvin! I'm so glad you liked the video! :)

[@cloudmichael11]

There is a formula for inhomogeneous linear ODEs of any order positive integer: For 2nd order: y_p = y_h*integral(y_h^-2(integral(W*y_h*e^(integral(P)dx)dx)*e^(integral(-P)dx)dx) so, the methods of undetermined coefficients & variation of parameters are obsolete. See my book available on amazon .

[@onlymelodies4038]

Thank you so much, this helped so much for my History exam on Friday!

[@chemiellee2881]

😭😭😭 I'm so hard in my heart you indeed save my life, God bless you and your family forever keep giving your generations abundant knowledge

[@itzdaman]

Because you would get an answer that would be wrong or make no sense. In this example you would get A-2C exp(-2x)+4C exp(-2x)=24x+exp(-2x), so here the solutions would be: A=0, C=1/2 and 24x=0 and the problem lies with the last one; 24*x=0 cannot be true because that means x=0 , buy x is your variable and not a constant, your solution should always hold for all x, not just 0.

[@kristakingmath]

Awesome!! So glad it helped, and GOOD LUCK on your finals!! :D

[@spencercarroll2346]

Thank you for taking the time to explain the particular solution process!!

[@kristakingmath]

You're welcome! Finding the particular solution can definitely be tricky. :)

[@kristakingmath]

I'm glad I can help along the way! :)

[@mbcx4av316]

this helped so much thank you! so hard to find youtube videos with the most difficult examples that i'm likely to get in an exam to practice!

[@kristakingmath]

I'm so glad it helped!

[@ouzairhafeez]

maybe tagging the equation, (for example like x+y+z=0 ---> Equation 1 ; and after finding the values we can say "Plugging the values now in Equation 1") would have been ittsy bittsy helpful in keeping track :) (just a thought). But what the heck another AWESOME Video, truly appreciate it and thank you so much for taking time out to make such amazing videos! :)

[@RobSummers993]

THANK YOU!!! ...I have no clue how you ever figured all of this out before there was a CalculusExpert on KZbin. lol :) Thanks for making this awesome channel!

[@kristakingmath]

+Robert Summers LOL, you're welcome, I'm so glad you're enjoying it!

[@robertsrozentals1022]

my university got closed because covid-19 and these kinda videos is really what l need to pass my exams they saved me thank you !!!!

[@kristakingmath]

You're welcome, roberts, I'm so glad the videos are helping! I hope you and your family are all staying safe and healthy! :)

[@mgofthefc]

I go to a "top university" but I am at the bottom of my class. I never understand what the teacher is talking about. You have essentially been my calculus teacher for the past 2 semesters. I would have been kicked out of school by now if it weren't for you! You do great work, if you had a "donate" button on your website, I would send you my bitcoins! I can't thank you enough.

[@Nadeenactyl]

man i wish i found you 4 months ago. you've helped me a lot, thanks!

[@kristakingmath]

Nadeen Abouseido You're welcome, I'm so glad the videos have been helping!

[@kristakingmath]

Wow, thank you for sharing! I'm so glad I was able to help along the way, but you still had to put in the work! :D

[@DanielRodrigues-nx5eu]

I apricciate all your calc videos. Thank you for help me getting throught calc 2 and 3. You're my second teacher.

[@kristakingmath]

I'm glad I can help!

[@Sherikovic]

Searched a lot for this example and finally found a helpful one, thank you!

[@kristakingmath]

Awesome! Glad you liked it. :)

[@ArrowWaters]

integralCALC Hey why do you have to add the general and particular solutions together...I don't see this on my mark scheme for a similar question, it just ends with the particular solution that was worked out.

[@joebiden382]

The cutest person to teach me math yet!

[@edwardomahaga9334]

My end term exam is saved. Thank you Krista!

[@kristakingmath]

Glad it could help!

[@kristakingmath]

I wish you had found it sooner as well! Hope your midterm still went well!

[@SNovaCobra]

10/10 saved my maths exam, you've earned a subscriber (:

[@innovate1983]

That was So good! Understood clearly first time around. Thanks

[@StopLossDill]

You do such a good job! I can't believe I'm only watching this now! Thanks so much, great job

[@kristakingmath]

Thank you so much, Dillan! I'm so glad it helped! :D

[@StopLossDill]

@@kristakingmath I had to rewrite an applied mathematics module with covid making it tricky the first time around. My immigration from South Africa to New Zealand was on the line and my Lecturer had been skipping details here and there. I eventually passed on that rewrite thanks to you!! Thank you so much!!! I’m in managed isolation in New Zealand right now!

[@RidinDutch37]

I have the worst calculus professor that can't explain sh*t...am I glad I found you!

[@GahtanWolf]

Thank you thank you thank you *1000000000000000 times! I'm taking def eq this summer and I was totally LOST ! You should be a college instructor. You make everything easy and understandable ! Greetings from UAE:)

[@kristakingmath]

shllak i'm so glad i was able to help! good luck with the rest of your Diff Eq class!

[@GolddiggerCG]

Thanks your videos are really helpful! exam tuesday... fingers crossed.

[@kristakingmath]

I'm happy I could help! :)

[@kristakingmath]

@MsKateC You're welcome! Glad I could help :)

[@kristakingmath]

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

[@kristakingmath]

i hope you did super great on your midterm!! :D

[@Kebab_with_extra_garlic_mayo]

9:17 why cant I say that the addition of those constants create a new random constant such that we don't have to multiply the Ax by another x? The same applies for just combining the coefficients of the exponents?

[@Fidonese]

Thank you so much. I had a similar question, but the e^(x) was multiplied by the polynomial. The line "just guess the polynomial, but make sure you include all the lower powers" got me the right answer. If you hadn't said that, I'd still be stuck. Thanks for paying attention to the smaller details.

[@kristakingmath]

Oh good, I'm so glad that helped! That's an easy step to overlook, but those lower powers are so important! :D

[@jacksoncollette2846]

thank you so much I've been struggling with particular solutions. haven't heard it put the way you worded it and it makes perfect sense now

[@kristakingmath]

I'm glad it could help! :D

[@alexwhb122]

Thank you very much for this video. I think you may have just saved me on a few of my test questions. :)

[@astralstriker]

Spellbound O_O u are really really awesome. only 2 days to my finals.. i was worried for 'em but not any more... Many Thanks!!!

[@shaheenhearts]

Seriouslllyyy! THANKYOU SOOO MUCH!!! i have an exam on monday and this helped soo much!!!!!!

[@themodernglory]

Thank you so very much ,like this was so helpful to me because i was once having a problem understanding differential equations then you came along lol, now everything looks so easy, you teach better than my calculus 4 professor. Your presentation, color , style and voice was so very helpful, Once again thanks for this video.

[@kristakingmath]

Thanks Ben! I'm glad the videos are helping!

[@Bosco6826]

Thanks with your video. I didnt listen to the lessons and it is 9 hrs left to my midterm test now haha.. your video saves me! I think I could get a pass haha

[@Prodigy906]

Thank you so very much! I am in college Calculus 2 and this helped me so much! I've been completely lost as he gave no clear steps and just muddied everything up with deriving all of the equations and proving them while teaching! I am still a bit confused on what we are actually trying to accomplish though, even though how to do it is cleared up a lot.

[@kristakingmath]

LOL, I'm so glad it helped!!

[@kristakingmath]

so glad I could help! :)

[@ibanezkav]

o my god i love you :D for 8 weeks of classes i hadn't a clue what was going on and in the space of 20 minutes I understand cheers buddy ha :{D

[@danielzhang3725]

This may be a stupid question. But say that, hypothetically, the general solution was in the form of y=C1.exp(rx) + C2.x. exp(rx) ie. having only 1 root instead of 2, does that mean yp would have to equal to A.x^2+B.x+C.x^2.exp(-2x) due to the fact that C.x.exp(-2x) would overlap with the C2.x.exp(rx) in the general solution?

[@biscu6780]

Best math teacher ever?

[@hessy222]

wow you have a lot of patient and concentration and the best part, love for math :D you teach me this in less than 20 min and I spent 3 hours leacture lost in world of wonder

[@kristakingmath]

I'm so glad I could help! :)

[@hessy222]

integralCALC thanks, I did my recitation today and it really helped. just a question, whats your suggestion on doing better in differential equation? I have problem answering new questions even though I know the concept :D thanks!

[@SULTAN007ist]

thank you so much.. I study this subject as a part of Mathematical Engineering course,one reason that makes me hate this course is that it formulas,feel like it Never ends!!!

[@booysenbafana4358]

Thank you so much... I've struggling with this for a week now. Thank you very much

[@kristakingmath]

So glad it could help!

[@henryigwe9111]

Thanks so much for making this explanation so simple to understand. I'm very grateful

[@kristakingmath]

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