Types of First-Order Differential Equation

Example of First-Order Differential Equation

First-Order Differential Equation Solution

First-order differential Equations are classified into several forms, each having its characteristics. Types of the First-Order Differential Equations:

Linear Differential Equations
Homogeneous Equations
Exact Equations
Separable Equations
Non-Linear First Order Differential Equations

Linear Differential Equation

A linear differential equation consists of a variable, its derivative, and additional functions. It’s expressed in the standard form as:

dy/dx + P(x)y = Q(x)

where,

P(x) and Q(x) may be functions of x or numerical constants.

Homogenous First Order Differential Equation

A homogeneous differential equation is a function of the form (f(x,y) \frac{dy}{dx} = g(x,y)), where the degree of (f) and (g) is the same. A function (F(x,y)) can be considered homogeneous if it satisfies the condition: (F(\lambda x, \lambda y) = \lambda^n F(x,y)) for any nonzero constant (\lambda).

In simpler terms, a differential equation is homogeneous if it involves a function and its derivatives in a way that maintains a consistent degree.

Example of Homogenous First Order Differential Equation

Consider the differential equation: (\frac{dy}{dx} = \frac{x^2 – y^2}{xy}). This equation is homogeneous because both the numerator and denominator have the same degree (1).

Exact Differential Equations

The formula Q (x,y) dy + P (x,y) dx = 0 is considered to be an exact differential equation if a function f of two variables, x and y, exists that has continuous partial derivatives and can be divided into the following categories.

The general solution of the equation is:

u(x, y) = C

Since, ux(x, y) = p(x, y) and uy (x, y) = Q(x, y)

where, C is Constant of Integration

Separable Differential Equations

Separable differential equations are a special type of differential equations where the variables involved can be separated to find the solution of the equation. Separable differential equations can be written in the form of:

dy/dx = f(x) × g(y) where x and y are the variables and are explicitly separated from each other.

Once the variables have been separated, it is simple to find the differential equation’s solution by integrating both sides of the equation. After the variables are separated, the separable differential equation

dy/dx = f(x) × g(y) is expressed as dy/g(y) = f(x) × dx

Non Linear First Order Differential Equation

Nonlinear first-order differential equations do not fit the linear form and can involve powers or products of y and its derivatives. For example:

dy/dx = y² – x

First Order Differential Equation

A first-order differential equation is a type of differential equation that involves derivatives of the first degree (first derivatives) of a function. It does not involve higher derivatives. It can generally be expressed in the form: dy/dx = f(x, y). Here, y is a function of x, and f(x, y) is a function that involves x and y.

It defined by an equation dy/dx = f (x, y) where x and y are two variables and f(x, y) are two functions. It is defined as a region in the xy plane. These types of equations have only the first derivative dy/dx so that the equation is of the first order and no higher-order derivatives exist.

Differential equations of first order is written as;

y’ = f (x, y)

(d/dx)y = f (x, y)

Let’s learn more about First-order Differential Equations, types, and examples of First-order Differential equations in detail below.