Mixing Salt and Water - First Order Differential Equations

My 200th Video! Thank you for your support. 6.5K subscribers and 1.7 million views as of December 10, 2018. My goal is to double that in 2019.
In this video, we use first order, linear, ordinary differential equations to solve a mixing problem. We have a 3000L tank that is being filled and drained at the same time at 10L/min.
The solution filling the tank has a salt concentration of 0.02 kg/L, while the tank has an initial quantity of salt of 15kg. Our problem is to find the amount of salt in the tank at any given time.
Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.
Please ask me a maths question by commenting below and I will try to help you in future videos.
I would really appreciate any small donation which will help me to help more math students of the world. Please donate here: paypal.me/Mast...
Follow me on Twitter! MasterWuMath

[@user-youung.hyun26]

Oh my Jesus, thank you teacher Finally I understood what I have to do in nixing problem. I appreciate it.

[@scottboyer7774]

This video explains the topic much better than my professor! Thank you for uploading this

[@MasterWuMathematics]

You're very welcome! Thank you.

[@suchithsunku1390]

Thanks for explaining a problem to me 3:00 in the morning really saved my butt!

[@MasterWuMathematics]

Wow, you should get a good nights sleep before an exam. But thank you!

[@steveying1305]

very good explanation, and i think this problem is indeed quite tricky

[@MasterWuMathematics]

Yes, you are right

[@ryantan8169]

very helpful, thanks. Im gonna show my mates now

[@MasterWuMathematics]

Please do!

[@ryantan8169]

@@MasterWuMathematics had my vce specialist maths exam last week and this was on it

[@pipertripp]

Nice presentation. Bonus points for plotting the tank concentration v time at the end. That was a nice way to wrap it up.

[@MasterWuMathematics]

Thanks 👍

[@bwtscience8670]

Great tutorial!!!❤

[@MasterWuMathematics]

Thank you!! 😊

[@sevenmaster7820]

Appreciate your hard work sir ❤

[@starbeoms]

why leave out /min? dy/dt = 0.2[kg/min] - y/300[kg/min] i think dt is time and /min too.

[@sahebOp_]

Thank u! Very helpful for JEE❤

[@anp9929]

jesus christ brother you;ve solved it

[@filipomarcellino1854]

thank you soo much this really helped me

[@MasterWuMathematics]

You're welcome! Glad to help!

[@felixxxhuang9639]

This rlly helps me out!

[@jakefingy9048]

Appreciate the help

[@AJ-et3vf]

Awesome video! Thank you!

[@MasterWuMathematics]

Thanks for watching!

[@latajamadar2499]

Nice explanation

[@nope2404]

Very helpful. Thanks so much

[@PElizabeth]

Thank you!!! This helped me a lot.

[@MasterWuMathematics]

You're welcome!

[@Johnny-rg6tv]

THANK YOU THANK YOU THANK YOU!!!!!!!!!!!!!!

[@rodericksibelius8472]

I don't understand how you got a '-1' under the 'ln (60 -y) / -1' ....

[@palesapolette8168]

Thank you so much

[@kaokaoylan4510]

THANKS , GREAT HELP FOR MY REVISIONS!!~

[@mtimez1636]

is there matlab solution for this?

[@afiqhaiqal3706]

Hi sir, what if the rate in and rate out is different? Can we use your method?

[@MasterWuMathematics]

Yes, you can. I'll upload a sequel video with an example soon.

[@afiqhaiqal3706]

@@MasterWuMathematics tq sir 🥺

[@MasterWuMathematics]

@@afiqhaiqal3706 My sequel video is up now: kzbin.info/www/bejne/lXWynYmAe7VqmZI

[@xyznihall]

thanks mate

[@SuperMtheory]

It seems that this process could be generalized. For example: The tank hold L liters of fluid, with K kg of salt, etc. Doing so would provide a general solution. I wonder how the general solution would look.

[@MasterWuMathematics]

You absolutely can generalise this! Have a go yourself, simply follow the same process as I have done in this video, but replacing the numbers with letters.

[@RefinerSimilitude]

I just flopped this part of the exam because I tried to include the existing salt in the tank with my rate out. Since I didn't do that correctly, I couldn't perform the next 2 follow up questions. Practice, practice, practice.

[@maalikserebryakov]

Mindless practising isn’t going to help. Mathematics doesn’t work like that. Practise only once you understand and know the problem. do you think newton just practised really hard until F=MA magically appeared in his brain?…lol. Mathematics is an intellectual process. It’s not like lifting weights. Anyway, you need to understand the purpose of a differential equation in general. Its in the name. DIFFERENTial equation. It is used to describe things where its too difficult to immediately build an equation for a quantity. But its easy to see how the quantity CHANGES. so you should never try to include the initial values (existing salt) in the differential equation because at that point its not even a differential equation and if you can do that why bother with the differential equation to begin with? Look only at how a variable CHANGES. When you solve the differential equation you will Then havr a general solution and it is the purpose of the general solution to accommodate _any_ type of initial value you had. In this case, you could sub in the starting salt ok i’m done with my schizo rant

[@pipertripp]

Quick question, how much nastier does the problem get if the volume of solution is changing with time as well? Say we started with 1500L of brine and then let the tank fill to a max volume of 3000L. Volume would be a function of time too so would be an interesting ODE. I can think of how I'd solve this numerically, but a look at an analytical solution would be interesting.

[@MasterWuMathematics]

See this video: kzbin.info/www/bejne/lXWynYmAe7VqmZI

[@sheldoncooper1988]

Well explained! Thank you!

[@mohammedail67]

عاشت ايدك

[@EEAMD-co6nw]

honestly thought it was JackFrags talking, a big gaming youtuber

[@hydr9756]

I loved how you demonstrated this concept! Thanks for the visual at the beginning.

[@MasterWuMathematics]

Glad you liked it!

[@stilljamming-oraangaataan-maan]

why is there is negative 1 (-1) on 5:35

[@randaya5854]

If you differentiate ln |60 - y|, you will multiply the derivative of the function by the function inside, which is -1. Hence, in reverse differentiation, -1 is used. I know it is 4 years old, but I can’t help it.

[@alistabasini403]

What if the rate in > rate out

[@MasterWuMathematics]

See this tutorial: kzbin.info/www/bejne/lXWynYmAe7VqmZI

[@hamadkhan4890]

i would not understand the graph clearly

[@MasterWuMathematics]

What is it that you do not understand?

[@ajonjon8725]

Like it, but got a message that I disliked the video. I dont know what happened...I Like it. Thank you

[@collinsmamadumetja1000]

A 1000-litre holding tank catches runoff from a chemical process. The tank initially holds 800 litres of water with 2 grams of pollution dissolved in it. Polluted water, containing 5 grams per litre of pollution, flows into the tank at a rate of 3 litres per hour. At the same time, the well-mixed solution leaves the tank at 3 litres per hour. When the amount of pollution in the holding tank reaches 500 grams, the inflow of polluted water is cut off and fresh water enters the tank at a decreased rate of 2 litres per hour, while the outflow is increased to 4 litres per hour. Determine the amount of pollution in the tank at any time 𝑡. HINT: Your answer will be a piecewise-defined function. Please solve for me.

[@thanyaniinnocent940]

60 is our asymptote, why is that so ?

[@gawain855]

Because the equation states y = 60 - 45e^(-t/300). As 45e^(-t/300) can never give a value lower than 0 (it's getting subtracted, so if it was negative it would be added to the 60), y can never be above 60.

[@cakefromkorea1507]

.

[@xkerrtv3496]

(Y/3000)x(10) = y/300 ? Wtf?

[@thomassellman1808]

it can also be seen as y*(1/3000)*10, which comes out to be y*(1/300) AKA y/300.