# First-order first-degree autonomous differential equation

## Contents

## Definition

Following the convention for autonomous differential equation, we denote the dependent variable by and independent variable by .

### Form of the differential equation

The differential equation is of the form:

### Solution method and formula

We convert the differential equation to an integration problem:

and carry out the integrations on both sides. If is an antiderivative of , the solution will be:

with the freely varying parameter over : every particular value of gives a solution. To express as a function of , we need to invert . If we can do so, we'd have:

as the general solution.

In addition, there may be stationary solutions. These are solutions that correspond to constant functions that satisfy .

## Related notions

- Separable differential equation: A slightly more general type of first-order differential equation.
- Second-order autonomous differential equation of degree one: Although such equations cannot always be solved, they
*can*always be reduced to first-order differential equations.

## Analysis

### Starting at time zero with value one

Suppose we want to solve the initial value problem for the differential equation:

subject to the initial condition that at , .

We consider various possibilities for a function that sends 1 and higher numbers to positive numbers, and make cases based on the growth rate of :

Nature of | Nature of in terms of ? |
---|---|

constant function | linear function |

(i.e., it grows roughly like a power function of ) | |

linear function | exponential function |

linear times logarithmic, something like | grows something like a doubly exponential function of (note: if we wanted growth between exponential and double exponential, we would need something like , and if we wanted triple exponential growth, we would multiply by a double logarithmic term) |

grows so fast in terms of that it reaches in finite time. |