First-order first-degree autonomous differential equation
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 .
- 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.
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.|