Introduction and First Definitions
- A differential equation is an equation involving an unknown
function and its derivatives.
- The order of the differential equation is the order of the
highest derivative of the unknown function involved in the equation.
- A linear differential equation of order n is a
differential equation written in the following form:
where is not the zero function. Note that some may use the
notation for the derivatives.
A linear equation obliges the unknown function y to have some
restrictions. Indeed, the only operations which are accepted for the
variable y are:
- Differentiating y;
- Multiplying y and its derivatives by a function of the
- Adding what you obtained in (ii) and let it be equal to a
function of x.
- Existence: Does a differential equation have a
- Uniqueness: Does a differential equation have more
than one solution? If yes, how can we find a solution which satisfies
- A problem in which we are looking for the unknown function
of a differential equation where the values of the unknown function
and its derivatives at some point are known is called an initial
value problem (in short IVP).
- If no initial conditions are given, we call the description of all
the differential equation the general solution.
[First Order D.E.]
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