1 Separating the variables in first order ODEs

In this Section we consider differential equations which can be written in the form

$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$

Note that the right-hand side is a product of a function of $x$ , and a function of $y$ . Examples of such equations are

$\frac{dy}{dx}=x^2{y}^3,\frac{dy}{dx}=y^{2}sinx\text{ and }\frac{dy}{dx}=ylnx$

Not all first order equations can be written in this form. For example, it is not possible to rewrite the equation

$\frac{dy}{dx}=x^2+y^3$

in the form

$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$

Task!

Determine which of the following differential equations can be written in the form

$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$

If possible, rewrite each equation in this form.

Answer

The variables involved in differential equations need not be $x$ and $y$ . Any symbols for variables may be used. Other first order differential equations are

$\frac{dz}{dt}=t{\text{e}}^z\frac{d\theta }{dt}=-\theta \text{ and }\frac{dv}{dr}=v\left(\frac{1}{r^2}\right)$

Given a differential equation in the form

$\frac{dy}{dx}=f\left(x\right)g\left(y\right)$

we can divide through by $g\left(y\right)$ to obtain

$\frac{1}{g\left(y\right)}\frac{dy}{dx}=f\left(x\right)$

If we now integrate both sides of this equation with respect to $x$ we obtain

$\int \; <!--nolimits\; -->\frac{1}{g\left(y\right)}\frac{dy}{dx}dx=\int \; <!--nolimits\; -->f\left(x\right)dx$

that is

$\int \; <!--nolimits\; -->\frac{1}{g\left(y\right)}dy=\int \; <!--nolimits\; -->f\left(x\right)dx$

We have separated the variables because the left-hand side contains only the variable $y$ , and the right-hand side contains only the variable $x$ . We can now try to integrate each side separately. If we can actually perform the required integrations we will obtain a relationship between $y$ and $x$ . Examples of this process are given in the next subsection.