Separation of Variables // Differential Equations

In this video we talk about our first major method for solving differential equations: the method of separation of variables. A first order ODE is separable if it can be written as y'=f(t)g(y), that is a product of two factors that each only depend on one of the variables. If that is the case, then we can separate and integrate. We're going to see in this video the first core example of exponential growth before jumping to a more complicated example which has an implicit solution and singular solutions

0:00 Exponential Growth
3:55 Separation of Variables
6:00 2nd Example
7:54 Singular Solution

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[loden]

I’ve taken calc 1 self study these last couple months and it’s gone pretty smoothly but currently doing separable differential equations has become quite the challenge! Your video here has helped me approach it from a fresh perspective! Thank you

[maximofernandez]

Man, you saved my life. I missed two classes of Mathematical Analisis II, and really had troubles following the things. Thank you so, so much for your content

[MrZWolfy]

You're contagious! In an awesome way! Thank you for your enthusiasm and keep it up. Wish more teachers & professors shared the same passion for their subjects.

[canadianmaple]

As a first year math student it's so interesting to see the many branches of mathematics. Pure math is still definitely more intimidating to me at this level than applied. Thank you for another awesome video!

[ted]

Taking an ODE's class at the exact same time as this series, nice videos to have to supplement the course, thank you!

[aselim.]

I watched it third time in 2024-2025 semester and I have taken notes.

[joshuaisemperor]

Great video as always Dr.! Must say again I love how you edit all the math onto the screen so clear!

[jmgpqcx]

Ah! I'm taking series and matrices this trimester and would have been so much excited to see any of that. Anyway, I'll come back for this series in the future.😄

[angelmendez-rivera]

At the risk of sounding pedantic, I want to start by clarifying some things.

Firstly, in the equation dy/dt = k·y, it is important to realize that, if you divide by y, then y must not be equal to 0 for any t, and since you cannot guarantee this a priori, you need to do some case analysis. If y = 0 on some interval, then dy/dt = k·0 = 0 on some nonempty open interval, and antidifferentiating thus implies y = C for some arbitrary C, for all t. Since y = 0 in some nonempty open interval, y = 0 for all t. This is one solution. With this, the remaining case is that there is no nonempty open interval where y = 0, hence you can safely divide by y, noting that y = 0 for all t is a solution.

Secondy, at the extent that you antidifferentiate 1/y with respect to y, you technically need to consider this in separate cases, y < 0, and y > 0, because, as the integrand has a singularity at y = 0, the domain is disjoint, so different constants of integration are implied for both parts of the domain. If y < 0, then this results in ln(–y) + A = k·t, which can be simplified to –y = e^(–A)·e^(k·t). Thus y = –e^(–A)·e^(k·t). –e^(–A) is just an arbitrary negative real constant, so you can just recycle the name and call it A, acknowledging that A < 0, so y = A·e^(k·t) is a solution family. If y > 0, then ln(y) + B = k·t, and an analogous argument resuls in y = B·e^(k·t), with B > 0. With this, the three solution families are y = A·e^(k·t) with A < 0, y = 0, y = B·e^(k·t). As it happens, all three families can be expressed as special cases of the single solution family y = C·e^(k·t) with arbitrary real C. This does give the same result in the video, but this is a more careful derivation of it.

Similar care should be taken in the general case dy/dt = f(t)·g(y)

[kreegory]

This guy is a legend. Single-handedly saving my math grades in college.

[ugestacoolie]

for some reason, our math teacher thought it was a good idea to just throw how to solve 2nd order differential equations at our face without even introducing the big underlying ideas, like why we can choose y = ae^bx as a general form, or why there are 2 parts to a non-homogeneous diff equation. Thank you for this course, and I will learn about them from the start

[VIKASVERMA-nkuw]

Plzz Make complete videos about Engineering mathematics sequencely . So from this I feel so glad to learn with u . Thank you very much

[tejasvibirdh]

Sir, I am your fan now... The way you teach is good and your excitement for teaching excites me to understand the concept...
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[tww]

Thank you for the reminder on exponent rules that results in ce^kt 3:08

[sarvasvkakkar]

Sir your explanation is really wonderful!!
I needed videos on this course as this is what is going to be taken up this semester in our college.

[georgesadler]

Professor Bazett, thank you for a great explanation on The Separation of Variables in Introduction Differential Equations. Calculus is a strong tool for understanding this and future topics in Differential Equations.

[peagyinpaul]

Been looking for the right video for the past two days and i finally got one. Thanks s lot Sir

[peterhindes]

At 3:25 isn't the reason we drop the absolute value because it is included in our new multiplicative C. Before that the abs(y) implied that we have two equations mirrored about the x axis.

[revathinarasimhan]

Nice explanation of the nature of the algebraic solution you found. I am teaching DE's this semester and will be following your videos, and adding links to your notes too ! My course is laid out pretty much along similar lines - with an emphasis of qualitative features of solutions.

[mashroorzisan]

Dr. Trefor you are doing great tutorial...i love your work. Keep making more videos.