First Order Differential Equations and Its Types: A first order differential equation is an equation that relates the derivative of an unknown function to the function itself. First order differential equation is an eqtn that involves only the 1st derivative of the function which is unknown. The general form of a first-order ordinary differential equation (ODE) is:

where represents the derivative of the unknown function with respect to the independent variable , and is a given function of both and .

The method of solving a particular differential equation depends on its type and can involve techniques such as separation of variables, integrating factors, substitutions, or using specific solution methods for certain types of equations.

First-order differential equations can be classified into different types, depending on the nature of the function . Some common types of first-order ODEs include:

Solve the differential equation

**Solution:**

To solve this equation, we can separate the variables and integrate both sides with respect to :

Integrating both sides:

where is the constant of integration.

Solve the differential equation

.

**Solution:**

This equation is a separable differential equation. We can separate the variables and integrate as follows:

Integrating both sides:

Taking the exponential of both sides:

Since the absolute value can be positive or negative, we can rewrite it as:

where indicates that the solution can take either the positive or negative form. We can simplify the expression further by combining the constant and the sign:

where is a nonzero constant.

**Initial Value Problem**

The answer (or its derivatives) must fulfil the initial condition(s), which is/are a collection of points.

**Examples:**

For a (DE) equation which involved in a function f(t). the initial conditions (ICs) are of the form:

A differential equation having initial conditions is an initial value problem, or IVP for short.

In this type, the equation can be rearranged in a way that separates the variables and , allowing them to be integrated separately. The general form is

, where and are functions of and , respectively.

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These equations have the form

, where and are given functions of . Linear differential equations can be solved using integrating factors or other methods.

In this type, the equation can be expressed in the form

, where is a function of the ratio . Homogeneous equations can be solved using a substitution such as .

These equations can be written in the form

, where and are functions of and . Exact equations can be solved by finding a function called the potential function or integrating factor.

These equations have the form

, where and are given functions of , and is a constant. Bernoulli equations can be transformed into linear equations by making an appropriate substitution.