theory for separation of variables
The first order (http://planetmath.org/ODE) ordinary differential equation^{} where one can separate the variables has the form where $\frac{dy}{dx}$ may be expressed as a product^{} or a quotient of two functions (http://planetmath.org/ProductOfFunctions), one of which depends only on $x$ and the other on $y$. Such an equation may be written e.g. as
$\frac{dy}{dx}}={\displaystyle \frac{Y(y)}{X(x)}}\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\displaystyle \frac{dx}{dy}}={\displaystyle \frac{X(x)}{Y(y)}}.$  (1) 
We notice first that if $Y(y)$ has real zeroes (http://planetmath.org/ZeroOfAFunction) ${y}_{1},{y}_{2},\mathrm{\dots}$, then the equation (1) has the constant solutions $y:={y}_{1},y:={y}_{2},\mathrm{\dots}$ and thus the lines $y={y}_{1},y={y}_{2},\mathrm{\dots}$ are integral curves. Similarly, if $X(x)$ has real zeroes ${x}_{1},{x}_{2},\mathrm{\dots}$, one has to include the lines $y={y}_{1},y={y}_{2},\mathrm{\dots}$ to the integral curves. All those lines the $xy$plane into the rectangular regions. One can obtain other integral curves only inside such regions where the derivative^{} $\frac{dy}{dx}$ attains real values.
Let $R$ be such a region, defined by
$$ 
and let us assume that the $X(x)$ and $Y(y)$ are real, continuous^{} and distinct from zero in $R$. We will show that any integral curve of the differential equation (1) is accessible^{} by two quadratures.
Let $\gamma $ be an integral curve passing through the point $({x}_{0},{y}_{0})$ of the region $R$. By the above assumptions^{}, the derivative $\frac{dy}{dx}$ maintains its sign on the curve $\gamma $ so long $\gamma $ is inside $R$, which is true on a neighbourhood $N$ of ${x}_{0}$, contained in $[a,b]$. This implies that as $x$ runs the interval $N$, it defines the ordinate $y$ of $\gamma $ uniquely as a monotonic function $y\mapsto y(x)$ which satisfies the equation (1):
$${y}^{\prime}(x)=\frac{Y(y(x))}{X(x)}$$ 
The last equation may be written
$\frac{{y}^{\prime}(x)}{Y(y(x))}}={\displaystyle \frac{1}{X(x)}}.$  (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
${\int}_{{x}_{0}}^{x}}{\displaystyle \frac{{y}^{\prime}(x)dx}{Y(y(x))}}={\displaystyle {\int}_{{x}_{0}}^{x}}{\displaystyle \frac{dx}{X(x)}}.$  (3) 
Because $y=y(x)$ is continuous and monotonic on the interval $N$, it can be taken as new variable of integration (http://planetmath.org/SubstitutionForIntegration) in the left hand side of (3): substitute $y(x):=y$, ${y}^{\prime}(x)dx:=dy$ and change the to $y({x}_{0})={y}_{0}$ and $y(x)=y$.

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Accordingly, the equality
$\int}_{{y}_{0}}^{y}}{\displaystyle \frac{dy}{Y(y)}}={\displaystyle {\int}_{{x}_{0}}^{x}}{\displaystyle \frac{dx}{X(x)$ (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 $R$.

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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 $\ne 0$, then one and only one integral curve of (1) passes through this point, the curve is regular (http://planetmath.org/RegularCurve), 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.

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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
$\int \frac{dy}{Y(y)}}={\displaystyle \int \frac{dx}{X(x)}},$ (5) which thus the general solution of the differential equation (1) in $R$. Hence one can speak of the separation of variables^{},
$\frac{dy}{Y(y)}}={\displaystyle \frac{dx}{X(x)}},$ (6) and integration of both sides.
References
 1 E. Lindelöf: Differentiali ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).
Title  theory for separation of variables 

Canonical name  TheoryForSeparationOfVariables 
Date of creation  20130322 18:37:43 
Last modified on  20130322 18:37:43 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  15 
Author  pahio (2872) 
Entry type  Topic 
Classification  msc 34A09 
Classification  msc 34A05 
Related topic  InverseFunctionTheorem 
Related topic  ODETypesReductibleToTheVariablesSeparableCase 