Separation of Variable - What's REALLY going on?

[MaxL]

So I've always done simple ODEs by the method of separation of variables. You know,

dy/dx = A*y
dy/y = A*dx
IndefiniteIntegral[1/y dy] = IndefiniteIntegral[A dx]
ln(y) = A*x + Constant
y = Constant*e^(A*x)

It's easy to remember and it usually works. A lot of the PDEs I know how to do involve this process at some point.

The problem is, all my professors were quick to point out that this is an abuse of notation. That is, dy/dx isn't really a fraction, and can't necessarily be treated as such--it just happens to work out well if you pretend in a simple ODE.

So what's the rigorous way to do this ODE? What's really going on when you use the 'pretend derivatives are fractions' trick?

 

[torquil]

When writing equations linear in terms lik dx, it is "shorthand" for expressions that are within integrals.

dy/dx = A*y

1/y * dy/dx = A

\int 1/y * (dy/dx) dx = A*\int dx

\int 1/y dy =A*\int dx

ln(y) = A*x + C

y = const*exp(A*x)

When the integral on the left is converted to an x-integral, I have used a change of integration variables from x to y. So it's the same, only more correctly written.

Torquil

 

[LCKurtz]

The type of DE which you can separate always has this form:

h(y)y' = g(x)

In separation of variables you write it as

h(y)dy = g(x)dx

Then you integrate the left side with respect to y and the right side with respect to x yielding

H(y) = G(x) + C

where H and G are the antiderivatives of h and g. Why indeed should this work, given that you did different things to opposite sides of the equation? Should be a no-no shouldn't it?

Let's instead integrate both sides with respect to x in the first place

$$\int h(y)y'\, dx = \int g(x)\, dx$$

Now, notice that if H is an antiderivative of h then:

$$\frac d {dx} H(y) = H'(y)\frac{dy}{dx}= h(y)y'$$

where the first step is the chain rule. So H(y) is an antiderivative on the left side and G(x) on the right and we get:

H(y) = G(x) + C.

That is what justifies the shortcut method of separation of variables. It gives the correct answer and this shows why.