First-order first-degree differential equation
Contents
Definition
In implicit form
A first-order first-degree differential equation is a differential equation that is both a first-order differential equation and a first-degree differential equation. Explicitly, it has the form:
where are known functions. Here,
is the independent variable and
is the dependent variable.
In explicit form
Any first-order first-degree differential equation can be converted to an (almost) equivalent first-order explicit differential equation, i.e., a differential equation of the form:
Conversion between the forms
A first-order first-degree differential equation can be converted to explicit form as follows. Start with:
Now, divide both sides by and set
, giving the explicit form.
Note that the process may involve some slight change in the set of solutions. In particular, any solution that identically satisfies both and
may be lost when we divide. In most cases, there are no such solutions, and there are usually at most finitely many such solutions.
Existence and uniqueness of solutions
- Peano existence theorem guarantees the existence of a local solution to any initial value problem for an explicit first-order first-degree differential equation
with initial value point
provided that
is continuous.
- Picard-Lindelof theorem establishes existence and uniqueness under somewhat stronger continuity and differentiability assumptions.
Solution strategies in particular cases
Below are some formats of equations for which general strategies are known. Note that the letter is no longer used for the solution function but may be used for other functions:
Equation type | Quick summary of solution strategy |
---|---|
first-order linear differential equation which in simplified form looks like ![]() |
Use the integrating factor ![]() ![]() ![]() |
separable differential equation which is of the form ![]() |
Separate and solve as ![]() ![]() ![]() |
first-order exact differential equation ![]() |
Try to find a relation ![]() ![]() ![]() |
Bernoulli differential equation ![]() ![]() |
Divide both sides by ![]() ![]() ![]() ![]() ![]() |