Question

How to solve an exact first order differential equation?

Answer 1

Exact Equations

A first‐order differential equation is one containing a first—but no higher—derivative of the unknown function. For virtually every such equation encountered in practice, the general solution will contain one arbitrary constant, that is, one parameter, so a first‐order IVP will contain one initial condition. There is no general method that solves every first‐order equation, but there are methods to solve particular types.

Gi

ven a function f( x, y) of two variables, its total differential df is defined by the equation

Example 1: If f( x, y) = x 2 y + 6 x – y 3, then

The equation f( x, y) = c gives the family of integral curves (that is, the solutions) of the differential equation

Therefore, if a differential equation has the form

for some function f( x, y), then it is automatically of the form df = 0, so the general solution is immediately given by f( x, y) = c. In this case,

is called an exact differential, and the differential equation (*) is called an exact equation. To determine whether a given differential equation

differential equation M dx + N dy = 0 is exact if and only if