How to determine if an differential equation is separable?
Any tips on determining if an differential equation is separable. Starting points etc. Also if an differential equation is separable how to go on and find a general equation for this.
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$\begingroup$Given an equation of the form $\frac{dy}{dx} = f(x,y)$, it is separable if and only if $f(x,y) = g(x)h(y)$, i.e., if your equation is of the form $\dfrac{dy}{dx} = g(x)h(y)$ on an open interval . If this is the case, the solution is found by integrating the following:
$$\int\frac{1}{h(y)}\frac{dy}{dx}\text{d}x = \int g(x)\,\text{d}x$$
Using the substitution $y$, $dy = \frac{dy}{dx}\text{d}x$, we can rewrite this solution in the more familiar form
$$\int\frac{1}{h(y)}\,\text{d}y = \int g(x)\,\text{d}x$$
Supposing $H(y)$ is an antiderivative of $\frac{1}{h(y)}$, $G(x)$ an antiderivative of $g(x)$, then the solution is given in the (in general, implicit) form $$H(y) = G(x) + C$$ for $C$ a constant of integration.
Replying to the comment, One thing that is usually omitted from explicit notation, but implied by the context in which such an equation appears, is that $y$ is being considered as an implicit function of $x$ in such an equation (one of many uses of the implicit function theorem in the subject of differential equations). As such, we should perhaps write $\frac{dy}{dx} = f(x,y(x)) = g(x)h(y(x))$, but this is often omitted when the context makes it clear that this is the intended meaning.
Also, while I have made no attempt to be completely rigorous in my construction, one thing worth clarifying is the fact that whenever we attempt to solve an equation like this, or even give a characterization of separability, we must consider the interval on which we can guarantee a solution exists. For instance, if we are looking at the initial value problem $$\frac{dy}{dx} = f(x,y) = g(x)h(y),\quad y(x_0)=y_0,$$ then we must appeal to the existence and uniqueness theorem; i.e., we must find a rectangle $R = \{(x,y)\,| \, a < x < b, c < y < d\}$, where $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continuous (in the sense of functions $\mathbb{R}^2\to \mathbb{R}$). Then, applying the existence and uniqueness theorem, we are able to conclude that for any $(x_0,y_0)$ in the rectangle $R$, there exists a unique (albeit perhaps implicit) function $y=y(x)$ satisfying the given differential equation, for $x$ in the interval $(x_0-\varepsilon,x_0+\varepsilon)$, for some (generally very small) $\varepsilon > 0$.
(Note that I do not claim that the existence and uniqueness theorem guarantees that we have a solution $y(x)$ for every $x$ in $(a,b)$. What it says is that if we start out at $(x_0,y_0)$ in this rectangle $R$, then we can guarantee that there is a unique solution passing through the point $(x_0,y_0)$ satisfying the equation, for $x$ in some open interval about $x_0$.)
Again, I was trying to motivate the method, but not necessarily give a fully rigorous construction of the result. Hopefully this clarifies things further.
$\endgroup$ 1 $\begingroup$A differential equation is separable if you can write it in the form $f(y) dy=g(x) dx$ . Then once you have it in this form just integrate both sides.
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