Justification for manipulations according to Leibniz-notation

Question

Is there a way to justify the manipulations according to Leibniz-notation without nonstandard-analysis.

E.g. $\frac{dy}{dx} = x \\ dy = x dx\\ \int dy = \int x dx\\ y = \frac{1}{2} x^2$

Answer 1

Since the OP included the tag nonstandard-analysis with his original question, it may be fair to point out that it is not quite accurate that the expression $\frac{dy}{dx}$ is just "notation". Indeed, Leibniz's view point was that $\frac{dy}{dx}$ is a ratio of infinitesimal differentials. Similarly, in the hyperreal framework it can be viewed as a ratio of infinitesimals, as explained in Keisler's textbook Elementary Calculus.

As far as the equation $\frac{dy}{dx}=x$ is concerned, this is not really necessary because a simple integration does the trick, but for example for $\frac{dy}{dx}=y$ we get a nontrivial example of separation of variables where a hyperreal framework can be useful in justifying the procedures usually followed in this approach.

Answer 2

There is actually no justification for multiplying both sides by $dx$ since $\frac{dy}{dx}$ is just a notation. However, there is a justification for integrating both sides with respect to x and the result of the left hand side is $y$ according to the Fundamental theorem of Calculus.

Answer 3

This thread showcases my main issue with the "Separation of Variables" that is taught in Differential Equations classes. There is no need for nonstandard analysis in Separation of Variables.

In class people are usually taught to manipulate differentials and are told its not rigorous but it works. What they are really using is the chain rule since: The chain rule states: $\frac{d}{dt}f(y(t)) = f'(y)\frac{dy}{dt}\\$ We can now use "Separation of Variables": $$ y'(t) = y\\ \frac{y'}{y}=1 \\ \text{denote: } \frac{1}{y}=f'(y)\\ f'(y)*y'=1\\ $$ Notice this is the same form as earlier meaning it can be integrated using the Fundamental Theorem of Calculus and integrating with respect to time $$ \int\frac{1}{y}y'dt=\int dt\\ \ln(y)=t+C\\ y=Ae^{t} $$

Thinking in terms of the chain rule removes the need for Leibniz notation and makes Separation of Variables Rigorous