❖ Exact Differential Equations ❖

📚 Exact Differential Equations: What They Are and How to Solve Them 🧮

In this video, I explain what it means for a differential equation to be exact and walk you through solving an example step-by-step. We begin by checking if the differential equation meets the necessary criteria, and then use partial derivatives and integration to find the solution.

🚀 What you’ll learn:

How to identify an exact differential equation.
The process of using partial derivatives to test for exactness.
How to integrate and solve the equation once it’s verified as exact.
Exact differential equations are significant because they represent a special type of differential equation that can be directly integrated to find the solution. This makes them particularly useful for solving real-world problems in fields such as physics and engineering, where systems are often described by functions whose differentials are exact, meaning they are path-independent, which simplifies the analysis of state variables and their interactions.

Key concepts of exact differential equations:

Direct integration: Unlike other types of differential equations, exact ones can be solved by straightforward integration since the differential is "exact" or "total," representing the total change of a function, regardless of the path taken.
State functions in thermodynamics: In thermodynamics, exact differentials are closely related to state functions, which depend solely on the system's current state and not the path used to reach that state.
Applications across fields: Exact differential equations are applied in various areas such as fluid mechanics, electrical circuits, and chemical reactions, where the system is often represented by a potential function.
Testing for exactness: To confirm that a differential equation is exact, we check if a specific condition involving the partial derivatives of the equation's coefficients holds true.
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Watch now to master exact differential equations and apply them to solve real-world problems!

#ExactDifferentialEquations #DifferentialEquations #MathTutorial #PartialDerivatives #Calculus #Mathematics #STEM #MathHelp #EngineeringMath #Physics #StateFunctions #Integration #EducationalContent

[patrickjmt]

Hi all! Wanna help a Youtube education OG? Please post comments, questions and anything else on your mind in the comment section! so, don’t forget to LIKE, THUMBS UP, and SUBSCRIBE! I’d appreciate it greatly as it helps me :)

[DiezelFuel]

Shout out my nigga PatJMT been saving my ass since Calc1

[haloshotz]

Been with you since my calc days im in diff eq and you have yet to fail me
jmt - stay with me 4evers lol

[SimoniDulanto]

1.5 hours worth of lecture summarized in 6 minutes.

[patrickjmt]

@Zadamanim that is just how i want it.
i tutored two girls for about a year before they realized they had been watching my videos. : )
one day, one of the girls was like: you sound just like this guy on the internet whose videos i watch. you even have the same name...

[samirfersobe]

Clear as water and so much faster than Khan Academy.

[joshpoe]

the lives you have saved are endless
thank you

[muhanad]

Thank you so so much @PatrickJMT I been watching your videos since I start college.
You saved me in PreCal, Cal 1 and Cal 2 and now in Differential Equations. God Bless you for every man.

[eytanbrodsky]

Videos like this one are what really... DIFFERENTIATE... Patrick from other math tutors.

[kkfh]

For anybody wondering like I was: the reason the exactness test works is because in an exact equation, M is the partial derivative with respect to x of F, and N is the partial derivative with respect to y of F. By the properties of symmetry of second partial derivatives, the partial derivative of M with respect to y should be the same as the partial derivative of N with respect to X, because F_xy = F_yx (this is a Cal 3 concept, and I'm taking Cal 3 and DEQ at the same time for maximum suffering, so I hadn't learned this yet)

[cakulha]

It's amazing how you can explain in 5 minutes something I spent a week trying to understand. Thanks!!!

[kareem]

you are the reason i passed calc 1, calc 2 and will be the reason i pass diff equations. you deserve a nobel prize i swear because however you are the reason every one in my uni passed their math courses. thank you so much

[walcXP]

Just five minutes to completely explain a thing that I've been struggling to understand from my lecture notes for the past 20 minutes.
Great thanks!

[benjaminsmus]

I love it when he said "it will simply go away" 3:20

[KaleahMoran]

you posted this video when i was 5…now i’m using it 15 years later…thank you so much for making the topic so clear!

[WasifKhizar]

Almost 12 years later and still saving lives. kudos to you bruh.

[CaseyJG]

I should be paying you, not Iowa State, for my diff. eq. class, and my taxes should be funding your salary, not my professor who makes 90 grand per and is... lets just say (as politely as possible) inefficient. I literally would not have made it through the first 2 weeks without your videos.

[csanchezcuba]

Patrick your tutorials are awesome, i will make sure to include your name in my degree certificate :)

[Lexyvil]

The use of colour helps a lot! You explained in 2 minutes what I my professor couldn't convey in 30-60 minutes. Thanks! Now I just need to re-watch this video once a day until I get a better grasp of what to do in chronological order. It's still hard to know whether to differentiate, or to integrate.

[patrickjmt]

@kazmanX super duper appreciated. if i got $.05 per view, i would be a super happy guy : )