Can you explain the theory behind separable differential equations?

[samwinnick]

I'm just starting my DE class, although I've been familiar with separable DEs for a while. Although they're (so far) pretty straight-forward to solve, I don't really understand the theory behind seperable DEs. In calc 1, it was stressed that dy/dx is NOT a fraction that can be "taken apart." Looking at the definition of the derivative, it's clear that you cannot rewrite the limit as one limit divided by another limit, because the denominator would be 0, breaking a limit law. It seems to me that the point of derivatives is that we have this indeterminate 0/0 form that, given the context of the original function, we can solve. It seems to me that separating the limit would be like saying that the dy doesn't depend on the dx. This seems like an "abuse of notation" to me, just like the one often used to "proof" the derivative chain rule. Can somebody please help me clear this up?

 

[voko]

A separable equation is $\frac {dy} {dx} = f(x)g(y)$. The trick you are concerned with transforms that as $\frac {dy} {g(y)} = f(x)dx$ and it is indeed questionable. However, it can be transformed as $\frac {1} {g(y)} \frac {dy} {dx} = \frac {y'(x)} {g(y(x))} = f(x)$, and then integrated $\int \frac {y'(x)} {g(y(x))} dx = \int f(x) dx$. The expression on the left hand side is integrated by substituting $z = y(x)$, when it becomes $\int \frac {dz} {g(z)}$ which is different from the "trick" $\frac {dy} {g(y)}$ only in notation. That's why the trick works.

 

[HallsofIvy]

The derivative is, indeed, not a fraction but it is defined as the limit of a fraction with the result that it has the properties of a fraction: That is if f(y) is a function of y and y= g(x) is a function of x, then we can write f(g(x)) and differentiate with respect to x to get the chain rule: df/dx= (df/dy)(dy/dx). You cannot "prove" that by simply saying "the 'dy's cancel", but you can prove it by going back before the limit, canceling in the "difference quotients" and then taking the limit.

It is in order to use that "fraction property" that differentials are defined in terms of the derivative, usually in a second semester Calculus course. If we have dy/dx= f(x)g(y), where "dy/dx" is the derivative, not a fraction, we can then write dy/g(y)= f(x)dx where "dy" and "dx" are now differentials, not derivatives.

 

[samwinnick]

Thanks, Voko. That's exactly what I was looking for. Now I just need to refresh myself on the mechanics of the substitution rule...

HallsofIvy, I think you found where the hole in my knowledge is. Both the chain rule and the substitution rule (for integration) are something I've kind of "accepted" up until this point, but differentials are really the source of my problem. I'll go learn about differentials now.

 

[blue_raver22]

Separable differential equations (DEs) are a type of first-order differential equation that can be written in the form of dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y, respectively. The key concept behind separable DEs is that the variables x and y can be separated and solved independently.

The theory behind separable DEs is based on the fundamental theorem of calculus, which states that the derivative of a function is equal to the integral of its derivative. This allows us to manipulate the differential equation by separating the variables and then integrating both sides. This process results in a solution that satisfies the original DE.

The reason why dy/dx cannot be treated as a fraction in the traditional sense is because it represents the instantaneous rate of change, rather than a ratio of two quantities. However, in the context of separable DEs, treating dy/dx as a fraction is simply a notational convenience that allows us to separate the variables and integrate.

In essence, separable DEs work by using the fundamental theorem of calculus to manipulate the equation and solve for the dependent variable y in terms of the independent variable x. This allows us to find a general solution to the DE, which can then be further refined by applying initial conditions or boundary conditions.

I hope this helps to clarify the theory behind separable DEs. As you continue your studies in DEs, you will encounter other techniques and methods for solving different types of DEs. Keep in mind that each approach has its own underlying theory and assumptions, and it is important to understand these concepts in order to apply them effectively. Good luck with your studies!