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

or

(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): $$y'(x) \;=\; \frac{Y(y(x))}{X(x)}$$ The last equation may be written

(2)

(3)

• Accordingly, the equality

  (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

  (5)

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

  (6)

  
  and integration of both sides.

Bibliography

1

E. LINDELÖF: Differentiali- ja integralilasku III 1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).