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I'm reading a textbook and it is talking about the solution to a differential equation $P(x,y) \text{d}x +Q(x,y)\text{d}y = 0$. But I find it hard to understand this equation. What does it mean? And what is defined to be a solution of the equation?

For example, my textbook claims that for an equation of the form $X(x)F(y)\text{d}x + Y(y)G(x) \text{d}y = 0$, if $G(a)=0$ then " the function $x = a$ is a solution to the equation." I really don't understand. Can anyone help ?

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$dx$ and $dy$ are differentials, which can be understood as relative displacements along the tangent line to a curve. So if you start out at a point $(x,y)$ on the curve and move an arbitrary distance along the tangent line, if the $x$ coordinate changes by $dx$ and the $y$ coordinate changes by $dy$, then the differential equation says $P(x,y) dx + Q(x,y) dy$ should be $0$. If that is true for all points on the curve, the curve is a solution of the differential equation. If you consider $x$ to be the independent variable and $y$ the dependent variable, so that on the curve $y$ is a certain function of $x$, then $\dfrac{dy}{dx}$ is the usual derivative of $y$ with respect to $x$, and the differential equation says $P(x,y) + Q(x,y) \dfrac{dy}{dx} = 0$. But might just as well think of $x$ as the dependent variable and $y$ the independent variable, and then the differential equation says $P(x,y) \dfrac{dx}{dy} + Q(x,y) = 0$. It's the same curve, but looked at from different directions ($y$ as a function of $x$, or $x$ as a function of $y$).

In your example, $dx =0$, $x=a$ is constant and the tangent line is vertical, so this one doesn't work with $x$ as independent variable, only with $y$ as independent variable. In these terms the differential equation is $$X(x) F(y) \dfrac{dx}{dy} + Y(y) G(x) = 0$$ which is satisfied when $x$ is the constant function $a$: the first term is $0$ because $dx/dy = 0$ and the second term is $0$ because $G(a)=0$.

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