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A differential equation may be defined as an equation that consists of the derivative of an unknown equation. The derivatives of the function helps to find the rate of change of a function. A differential equation is an equation that relates the derivatives with the other functions. Differential equations are mostly helpful in the fields of biology, physics, engineering, and many more. Studying the solutions that satisfy the given equations and the properties of solutions is one of the main purposes of the differential equations.
What are the Differential Equations?
A differential equation can be defined as the equation that consists of at least one derivative of an unknown function, either an ordinary or a partial derivative. Suppose, the rate of change of a function y concerning x is inversely proportional to y, we express it as dy/dx = k/y.
A Differential equation can also be defined as an equation that includes the derivative (derivatives) of the dependent variable concerning the independent variable (variables) in calculus. The derivative helps to represent a rate of change, and the differential equation helps us represent a relationship between the changing quantity concerning the change in another quantity. Let y=f(x) be a function where y is a dependent variable, f is an unknown function, x is an independent variable.
Order of Differential Equations
The order of a differential equation helps to find the highest order of the derivative appearing in the given equation. Depending on the order, there are two types of differential equations
First-order differential equation
Second-order differential equation
Degree of Differential Equations
If we can express a differential equation in the terms of a polynomial form, then the integral power of the highest order derivative that appears is called the degree of the differential equation. The power of the highest order derivative given in the equation is the degree of the differential equation. To calculate the degree of the differential equation, we will require a positive integer as the index of each derivative. If a differential equation cannot be expressed in the terms of a polynomial equation having the same highest-order derivative as the leading term, then, the degree of the differential equation is not defined.
FAQs on RD Sharma Class 12 Solutions Chapter 22 - Differential Equations (Ex 22.10) Exercise 22.10 - Free PDF
1. What are the types of differential equations?
The Differential Equations can be Classified as Follows:
Ordinary Differential Equation
Partial Differential Equation
Homogeneous Differential Equation
Non-Homogeneous Differential Equation
2. What is an ordinary differential equation?
The “Ordinary Differential Equation”, which is also known as ODE, is an equation that contains only one single independent variable and one or more of its derivatives concerning the variable. Hence, the ordinary differential equation is represented as the relation having the independent variable x, the real dependent variable y, with some of its derivatives y’, y”, ….yn,… concerning x. The ordinary differential equation can be homogeneous or non-homogeneous which are discussed later.
3. What is a partial differential equation?
The equation that involves only partial derivatives of one or more functions of two or more independent variables is called a partial differential equation which is also known as PDE. Example:
δu/ dx + δ/dy = 0,
δ2u/δx2 + δ2u/δx2 = 0
4.What is a homogeneous differential equation?
A differential equation in which the degree of all of the terms is the same is known as a homogeneous differential equation. They can be written in the form of P(x,y)dx + Q(x,y)dy = 0, where P(x,y) and Q(x,y) are homogeneous functions of the same degree in general.
5. What is a non-homogeneous differential equation?
A differential equation in which the degree of all of the terms is not the same is known as a homogeneous differential equation. One of the types of non-homogeneous differential equations is the linear differential equation, which is similar to the linear equation. The differential equation which is given as (dy/dx) + Py = Q (Where P and Q are functions of x) is known as a linear differential equation. (dy/dx) + Py = Q (Where P, Q are constant or functions of y). The general solution of a non-homogeneous equation is y × (I.F.) = ∫Q(I.F.)dx + c where, I.F(integrating factor) = e∫pdx