For solving differential equations, especially the ones of the form
$$g(x)dx = h(y)dy$$
we solve the equation by integrating on both sides to reveal the solution.
Understanding this for differentiating the equation on both sides is relatively easy. We know that we can formulate an alternative equation in terms of differentials for the original equation involved and come out with a new differential equation that holds because of the properties of the differentials.
But how does it work for integration on both sides? Am I missing any point here? I have referred to multiple books but none give a satisfactory explanation. Integrating an equation on both sides seems really wrong, if I may dare to use the word.
Please help. I'm stuck with this thing and I can only begin to understand differential equations once this is cleared from my head.
Thank you very much!
Best Answer
The original equation was presumably $$h(y)\frac{dy}{dx}=g(x),$$ or something equivalent to this.
You are given the mysterious rule about "splitting" $\frac{dy}{dx}$. You probably were told at one time that $\frac{dy}{dx}$ is not a fraction, and now all of a sudden we are treating it as a fraction!
So let us not split it. Suppose that $H(y)$ is an antiderivative of $h(y)$, that is, a function whose derivative wiith respect to $y$ is $h(y)$. Let $G(x)$ be a function whose derivative with respect to $x$ is $g(x)$.
We recognize $h(y)\frac{dy}{dx}$ as the derivative with respect to $x$ of $H(y)$ (Chain Rule). So our equation can be written as $$\frac{d}{dx} H(y)=\frac{d}{dx}G(x).$$
Thus $H(y)$ and $G(x)$ have the same derivative with respect to $x$. So they differ by a constant, and we find $$H(y)=G(x)+C.$$
Now the important part: this is exactly what we get when we "split" $\frac{dy}{dx}$ and integrate on both sides. So whether or not the splitting and integrating makes sense, it gives the right answer.
If you wish, splitting and integrating can be treated as a senseless mnemonic that works, a "shortcut" to the real calculation using the Chain Rule. In fact, the individual terms $dy$ and $dx$ can be given meaning, but it is a little complicated. And Applied (and less Applied) people have an essentially correct intuition based on adding up "infinitely small" quantities. Unfortunately, it takes considerable effort to make that intuition rigorous.