Exact Equations Example 1

First example of solving an exact differential equation.

Category:

Education

Tags:

• exact
• differential
• equations

[gonzoredux]

i thought there was a critter in my house for a second....ha!

[ryuusel]

Wait, didn't see that you corrected it.

I should start watching the whole video before asking questions.

[gdullard]

great teacher, better comedian haha 3:09

[BannedLol4l]

@nejtilsvamp There may be cases (such as this one) where it is impossible to solve explicitly for y in terms of x, and so the implict solution is the best you can do.

If Psi turned out to be, say, x+y, then you could have said y = C - x as the solution. But if you get y*sin(x) + x^2*e^y - y = C, you can't find an expression like y = f(x), because there is no way to isolate y.

[4and6stringer]

@UniverselGamer

A2. Yes. Integrate psi y then add f(x). Get the derivative wrt x. Equate it with psi x to get the value of f'(x) to get f(x).

[4and6stringer]

@UniverselGamer

A1. psi might have terms that 1.) have y 2.) have y plus constant or 3.) or just constants only (which is what we got) When deriving psi wrt x only, all of these are treated as constants, thus all having derivative of 0. To be safe, you always assume 1.) or 2.) or both is true, that's why you put f(y) instead of C. If f(y) ends up as a constant then you have no problem there.

[UniverselGamer]

Q1. from "4:40" to "5:30", i don't understand why +C is replaced with f(y). And also i dont understand how when taking the partial derivative of f(y) will equal to 0 wrt x, if y is a constant wouldn't it still be f(y)?

Q2. if we took the antiderivative of psi(y) would that "+C" be replaced by f(x)?

[tam3ree]

Hey Sal, shouldn't you put "+C" at the end of psi at 6:40 ? Or is it included in f(y) ? Thank you!

[Jawshooah]

I'm always astounded at how much more fluidly and coherently Sal is able to explain mathematical and scientific concepts than my professors and TAs. You, sir, are awesome :D

[shmittythepirate]

@manny34711 yeah...

[manny34711]

shouldnt f'(y)=-1 instead of 1?

[Marinecorps1812]

@nejtilsvampe

I know you asked this a long time ago, but Psi is an intermediate step to solving the Differential Equation in terms of x and y.

Yes, it's preferred to solve for y explicitly, but sometimes it's hard so an implicit solution is okay.

For example: Y = X + 1 is the same as 1 = Y - X