First order and first degree differential equations and their geometrical interpretations

A first order and first degree differential equation involves the independent variable x (say), dependent variable y (say) so, it can be put in any one of the following forms:

dy/ dx = f(x, y) or f (x, y) = 0, or f(x, y) dx + g(x, y)dy = 0

Where f(x, y) and g(x, y) are functions of x and y.

Geometrical interpretation

The general from of a first order and first degree differential equation is f(x, y, dy/dx) = 0 … (i)

We know that the tangent of the direction of a curve in Cartesian rectangular coordinates at any point is given by dy/dx, so the equation in (i) can be known as an equation which establishes the relationship between the coordinates of a point and the slope of the tangent i.e., dy/dx to the integral curve at that point. Solving the differential equation given by (i) means finding those curves for which the direction of tangent at each point coincides with the direction of the field. All the curves represented by the general solution when taken together will give the locus of the differential equation. Since there is one arbitrary constant in the general solution of the equation of first order, the locus of the equation can be said to be made up of single infinity of curves.