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 ${\displaystyle \frac{dy}{dx}}$ 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

${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{Y(y)}{X(x)}}\mathit{\hspace{1em}}\text{or}\mathit{\hspace{1em}}{\displaystyle \frac{dx}{dy}}={\displaystyle \frac{X(x)}{Y(y)}}.$(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 ${\displaystyle \frac{dy}{dx}}$ 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 ${\displaystyle \frac{dy}{dx}}$ 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

${\displaystyle \frac{y^{\prime }(x)}{Y(y(x))}}={\displaystyle \frac{1}{X(x)}}.$(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

${\displaystyle {\int }_{x_0}^x}{\displaystyle \frac{y^{\prime }(x)dx}{Y(y(x))}}={\displaystyle {\int }_{x_0}^x}{\displaystyle \frac{dx}{X(x)}}.$(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  $y(x)=y$.

• Accordingly, the equality

  ${\displaystyle {\int }_{y_0}^y}{\displaystyle \frac{dy}{Y(y)}}={\displaystyle {\int }_{x_0}^x}{\displaystyle \frac{dx}{X(x)}}$(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 $R$.
  

• 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.

• 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

  ${\displaystyle \int \frac{dy}{Y(y)}}={\displaystyle \int \frac{dx}{X(x)}},$(5)

  which thus the general solution of the differential equation (1) in $R$.  Hence one can speak of the separation of variables,

  ${\displaystyle \frac{dy}{Y(y)}}={\displaystyle \frac{dx}{X(x)}},$(6)

  and integration of both sides.