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Mathematics 77 Online
OpenStudy (anonymous):

how to solve: y'(x)=e^(y/2)sinx ans: y=-2ln(1/2cosx+C)

OpenStudy (accessdenied):

This appeares to be a separable Differential Equation. Do you know how these generally play out? Like, what we should be looking to do?

OpenStudy (anonymous):

nope, my teacher is awful/I slept through class. So, I'm looking through my book for relevant examples, but there's not many.

OpenStudy (accessdenied):

Ah. Okay. We can tell that it is "separable" when we have the ability to move all y terms to one-side and the x terms to the other. Here's a really general example that basically gives a framework for any Separable differential equation. \( \displaystyle y'(x) = G(y) F(x) \) F(x) a general function of x and G(y) a function of y We use a bit of a trick here, using the idea that: \( \displaystyle y' = \frac{\text{d}y}{\text{d}x} \) Now we rwrite: \( \displaystyle \frac{\text{d}y}{\text{d}x} = G(y) F(x) \) Let's multiply both sides by "dx" and divide both sides by G(y). \( \displaystyle \frac{1}{G(y)} \; \text{d}y = F(x) \; \text{d}x \) Friom here, our solution method is simply integratingth sides. This will usually give us a general solution (sometimes y won't be right there in front of us), so we may have to do a little further Algebra depending on the G(y).

OpenStudy (accessdenied):

In our case, we have: y'(x)=e^(y/2)sinx G(y) = e^(y/2), F(x) = sin x So it follows the same format as above. \( \displaystyle y'(x) = e^{y/2} \sin x \) <-- lets change out y' = dy/dx \( \displaystyle \frac{\text{d}y}{\text{d}x} = e^{y/2} \sin x \) <--- "separate" moving dx & e^(y/2) to the correct sides. \( \displaystyle e^{-y/2} \; \text{d}y = \sin x \; \text{d}x \) <-- Here, we can integrate both sides. \( \displaystyle \int e^{-y/2} \; \text{d}y = \int \sin x \; \text{d}x \)

OpenStudy (accessdenied):

So far, do you see how that works out here?

OpenStudy (anonymous):

yes, I think I've got it I'll ask more question as I finish the problem.

OpenStudy (accessdenied):

Alright! :)

OpenStudy (accessdenied):

The situation with your teacher is unfortunate; from my experience, Differential Equations was a /lot/ of 'tricks' and things I would never have considered coming in from Calc I/II. The next topic following Separable Differential Equation is usually Linear DE, and one of the methods of solving them is using the property of derivatives that f' g + fg' = (fg)' and multiplying by a specific function to make that property work out / allow you to integrate both sides to break y out of the derivative.

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