When an equation has one or more functions as well as its derivatives, it is known as a differential equation. There are certain terms that we need to take care of while studying differential equations. Those terms are given below:
1. Dependent Variable: When an equation has only one variable, it is called a dependent variable.
2. Independent Variable: The dependent variable is dependent on another variable, which is known as the independent variable.
Note: The differential equations may or may not have either one or more than one dependent or independent variable.
Differential Equations have many use cases. They are taken into consideration almost in every field, be it chemistry, mathematics, biology, physics, engineering, and so on. From species of any living organism to rough engineering, chemical decomposition, population, and other areas of research, differential equations play a massive role.
The dependent variables, with consideration of independent variables, when forming a function derivative, then this phenomenon is known as differential equations.
Order and degree are the main terms that ought to be perceived while tackling the differential conditions. Order of a differential condition is the most noteworthy capacity to which the subordinates are brought up in the given condition. Be that as it may, degree then again is the force of the greatest subordinate. For instance, consider the differential condition referenced underneath.
(y’)2 + y’’’ - 2 (y’’)4 = 7y
The equation comprises the third derivative of 'y' as y''' which is the most noteworthy derivative. The power of y''' is 1. Thus, the level of the equation is '1'. Be that as it may, the subsequent derivative is y'' which is raised to power 4 which is the most elevated power of the derivative. Along these lines, the request for the given differential equation is 4.
The Different Types of Differential Equations
There are many different types of differential equations, starting with the basis of the type of variables, the types are:
1. Partial Differential Equations: When two or more two independent variables affect the dependent variable.
2. Ordinary Differential Equations: This generally depends on only one independent variable.
Types of Differential Equations Based on the Order of Equations:
1. First order of differential equation: When 1 is the highest power of the formed derivatives.
2. Second-order of the differential equation: When 2 is the highest power of the formed derivatives.
3. N(th) order of differential equation: When 'N' is the highest power of the formed derivatives.
Types of differential equations based on homogeneity:
Homogenous differential equations
Non-homogeneous differential equations
Solution of Differential Equations:
Solving a differential equation means finding an equation that does not contain any derivatives. However, this equation should satisfy the differential equation that is being solved. Solving differential equations involves two or more integrations. To determine an appropriate method to solve the differential equation, it is very important to identify the type of differential equation that is being solved. Both general and particular solutions of differential equations can be obtained by using appropriate steps to solve the equation.