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What are Differential Equations?
A differential equation is an equation that has a function, such as F(x), and one or even more derivatives, such as dy/dx. The physical quantities are commonly represented by the functions of a differential equation, while the rate of change of the physical quantities is given by the derivatives. A differential equation is a mathematical expression that describes the connection here between function and also its derivatives. When we examine y as a function of x, a differential equation is one that incorporates the derivatives of y with respect to x (or the differentials of y and x) with or without variables x and y.
Order of Differential Equations
It represents the highest order of the derivative. Here are some of the examples
First Order Differential Equation
The example of a first-order differential equation is given below. There is only one first derivative dy/ dx present.
For example – (dy/dx) = tan x
The first order derivatives are represented by
dy/dx =f(x,y)= y’
Second Order Differential Equation
Here is an example of a second-order differential equation. It is represented as :
Degree of Differential Equations
- The highest-order derivatives power is called the degree of the differential equation.
- The differential equation may be in the polynomial form in that the highest order of differential power is called a degree.
- Choose the positive integer of the find about the degree.
Types of Differential Equation
There are two types of differential equations. They are
- Ordinary differential equation
- Partial differential equation
Ordinary Differential Equations
- ODE – A differential equation comprising one or even more functions from one independent variable with their derivatives is known as an ordinary differential equation (ODE).
For example here are some derivatives y’+y”+y”’+y””+……… y^n concerning x
- The ordinary differential equation is further classified into two. They are
Example: (d2y/dx2) + (dy/dx) = 3y tan x
- Homogeneous differential equation
- Nonhomogeneous differential equation
Homogeneous Differential Equation
- The equation that contains differentiation, set of variables and function(x, y) is called a homogeneous differential equation.
- The homogeneous differential equation should have the same power for the given variables (x, y)
Homogeneous differential equation
f(x,y).dy + g(x,y).dx = 0
where x and y are variable having same degree
dx/dy = F(x,y)
Homogeneous function
f (δx,δy) = δ n f(x,y)
where δ is non zero constant
- There is no constant term present in the homogeneous differential equation. Only in the linear differential equation, you can see the constant term
- If we remove the constant term from the linear differential equation, then the equation would turn into the homogeneous differential equation
- No variable is present in the special functions like logarithm of trigonometric
Here are some example
You can substitute x and y in all the above examples to prove the homogeneous differential equation.
- x = δx
- y = δy
Substitute x/y = v or x= vy when the homogeneous differential equation is in the form of dx/dy=f(x,y) and has the homogeneous function f(x,y).
Non- Homogenous Differential Equation
It is much similar to that of a linear equation and the order of the differential equation is not similar.
For example, the differential equation of the form (dy/dx) + Py = Q where P, Q are the constant or the function of x
The general solution is given as
Y * ( integrating factors) = Q* (integrating factors).dx + c
where integrating factors = epdx
Formation of Partial Differential Equations
For more than one function and more than one independent variable is used. It is called Partial differential equation(PDE)
Here are a few examples that are
- 5𝛿u/ dx + 7𝛿u/𝛿xy = 0,
- 21𝛿2u/𝛿x2 + 8𝛿2u/𝛿x2 = 0
Conclusion :
The connection between both the variables x and y that is produced after eliminating the derivatives (i.e., integration) or where the connection includes an arbitrary constant to signify the order of an equation is the basic solution of a differential equation. An arbitrary constant appears in the solution of first-order differential equations, while two arbitrary constants appear in the solutions of second-order difference equations. The solution of the differential equations is derived by giving specific values to the arbitrary constant.
