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Hometheory for separation of variables
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theory for separation of variables
The first order ordinary differential equation where one can separate the variables has the form where $\displaystyle\frac{dy}{dx}$ may be expressed as a product or a quotient of two functions, one of which depends only on $x$ and the other on $y$. Such an equation may be written e.g. as
$\displaystyle\frac{dy}{dx}\;=\;\frac{Y(y)}{X(x)}\quad\mbox{or}\quad\frac{dx}{% dy}\;=\;\frac{X(x)}{Y(y)}.$  (1) 
We notice first that if $Y(y)$ has real zeroes $y_{1},\,y_{2},\,\ldots$, then the equation (1) has the constant solutions $y:=y_{1},\;y:=y_{2},\;\ldots$ and thus the lines $y=y_{1},\;y=y_{2},\;\ldots$ are integral curves. Similarly, if $X(x)$ has real zeroes $x_{1},\,x_{2},\,\ldots$, one has to include the lines $y=y_{1},\;y=y_{2},\;\ldots$ to the integral curves. All those lines divide the $xy$plane into the rectangular regions. One can obtain other integral curves only inside such regions where the derivative $\displaystyle\frac{dy}{dx}$ attains real values.
Let $R$ be such a region, defined by
$a<x<b,\quad c<y<d,$ 
and let us assume that the $X(x)$ and $Y(y)$ are real, continuous and distinct from zero in $R$. We will show that any integral curve of the differential equation (1) is accessible by two quadratures.
Let $\gamma$ be an integral curve passing through the point $(x_{0},\,y_{0})$ of the region $R$. By the above assumptions, the derivative $\displaystyle\frac{dy}{dx}$ maintains its sign on the curve $\gamma$ so long $\gamma$ is inside $R$, which is true on a neighbourhood $N$ of $x_{0}$, contained in $[a,\,b]$. This implies that as $x$ runs the interval $N$, it defines the ordinate $y$ of $\gamma$ uniquely as a monotonic function $y\mapsto y(x)$ which satisfies the equation (1):
$y^{{\prime}}(x)\;=\;\frac{Y(y(x))}{X(x)}$ 
The last equation may be written
$\displaystyle\frac{y^{{\prime}}(x)}{Y(y(x))}\,=\,\frac{1}{X(x)}.$  (2) 
Since $X$ and $Y$ don’t vanish in $R$, the denominators $Y(y(x))$ and $X(x)$ are distinct from 0 on the interval $N$. Therefore one can integrate both sides of (2) from $x_{0}$ to an arbitrary value $x$ on $N$, getting
$\displaystyle\int_{{x_{0}}}^{x}\frac{y^{{\prime}}(x)\,dx}{Y(y(x))}\,=\,\int_{{% x_{0}}}^{x}\frac{dx}{X(x)}.$  (3) 
Because $y=y(x)$ is continuous and monotonic on the interval $N$, it can be taken as new variable of integration in the left hand side of (3): substitute $y(x):=y$, $y^{{\prime}}(x)\,dx:=dy$ and change the limits to $y(x_{0})=y_{0}$ and $y(x)=y$.

Accordingly, the equality
$\displaystyle\int_{{y_{0}}}^{y}\frac{dy}{Y(y)}\;=\;\int_{{x_{0}}}^{x}\frac{dx}% {X(x)}$ (4) is valid, meaning that if an integral curve of (1) passes through the point $(x_{0},\,y_{0})$, the integral curve is represented by the equation (4) as long as the curve is inside the region $R$.

Additionally, it is possible to justificate that if $(x_{0},\,y_{0})$ is an interior point of a region $R$ where $X(x)$ and $Y(y)$ are real, continuous and $\neq 0$, then one and only one integral curve of (1) passes through this point, the curve is regular, and both $x$ and $y$ are monotonic on it. N.B., the Lipschitz condition for the right hand side of (1) is not necessary for the justification.

When the point $(x_{0},\,y_{0})$ changes in the region $R$, (4) gives a family of integral curves which cover the region once. The equations of these curves may be unified to the form
$\displaystyle\int\frac{dy}{Y(y)}\;=\;\int\frac{dx}{X(x)},$ (5) which thus represents the general solution of the differential equation (1) in $R$. Hence one can speak of the separation of variables,
$\displaystyle\frac{dy}{Y(y)}\;=\;\frac{dx}{X(x)},$ (6) and integration of both sides.
References
 1 E. Lindelöf: Differentiali ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Mathematics Subject Classification
34A09 no label found34A05 no label found Forums
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