I see an equation like this:

y\frac{\textrm{d}y}{\textrm{d}x} = e^x

and solve it by “separating variables” like this:

y\textrm{d}y = e^x\textrm{d}x

\int y\textrm{d}y = \int e^x\textrm{d}x

y^2/2 = e^x + cWhat am I doing when I solve an equation this way? Because \textrm{d}y/\textrm{d}x actually means

\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}

they are not really separate entities I can multiply around algebraically.

I can check the solution when I’m done this procedure, and I’ve never run into problems with it. Nonetheless, what is the justification behind it?

What I thought of to do in this particular case is write

\int y \frac{\textrm{d}y}{\textrm{d}x}\textrm{d}x = \int e^x\textrm{d}x

\int \frac{\textrm{d}}{\textrm{d}x}(y^2/2)\textrm{d}x = e^x + cthen by the fundamental theorem of calculus

y^2/2 = e^x + c

Is this correct? Will such a procedure work every time I can find a way to separate variables?

**Answer**

The basic justification is that integration by substitution works, which in turn is justified by the chain rule and the fundamental theorem of calculus.

More specifically, suppose you have: \frac{dy}{dx} = g(x) h(y)

Rewrite as:

\frac{1}{h(y)} \frac{dy}{dx} = g(x) Add the implicit dependency of y on x to obtain

\frac{1}{h(y(x))} \frac{dy}{dx} = g(x)

Now, integrate both sides with respect to x:

\int \frac{1}{h(y(x))} \frac{dy}{dx} \, dx = \int g(x) \, dx If we do a variable substitution of y for x on the left-hand side (i.e., use the integration by substitution technique), we replace \frac{dy}{dx} dx with dy. Thus we have \int \frac{1}{h(y)}\, dy = \int g(x) \, dx,

which is the separation of variables formula.

So if you believe integration by substitution, then separation of variables is valid.

**Attribution***Source : Link , Question Author : Mark Eichenlaub , Answer Author : Mutantoe*