Let’s begin by defining a differential equation first-
An equation comprising one or more variables (unknowns) and some of their derivatives is a differential equation. The differential equation is the bridge between the variables and the derivatives of those variables in itself. The order of an equation is determined by the order of the derivative with the highest possible order that it includes. The degree of a differential equation can be defined by the power to which the derivative of the highest order is raised.
Order
The order of an equation’s derivative with the highest order tells us what order the equation is in. The exponent of the derivative with the highest order in a differential equation is what defines the degree of the equation. All of the derivatives in the equation, both positive and negative, have no fractional powers. None of the fractions have anything to do with derivatives.
If the differential equation doesn’t meet one or more of the above conditions, it should first be changed into a form that does meet all of the above conditions. If an equation cannot be reduced, then it is said to have no degree or an undefined degree.
Differential equations are put into groups based on the order in which they happen. In a differential equation, the order of the equation is determined by the derivative that has the highest power. If a function and all of its derivatives are raised to the first power, and if the coefficient of each derivative in the equation only employs the independent variable x, then we say that the equation is linear. A property of first-degree equations is linearity.
First Order Differential Equation
A first-order differential equation can be defined by the equation dy/dx =f (x,y) where x and y are variable and a function f(x,y) can be defined on a region in the xy-plane. It only has the first derivative, dy/dx, therefore it can be said that it’s a first-order equation where no higher-order derivatives are present.
The first-order linear equation can be written in a reduced form as
(dy/dx) + Py = Q
P and Q are either constant or only depend on the variable that is not being changed. This represents a first order linear differential equation.
A first-order differential equation is defined as dy/dx =f (x,y) with f(x,y) defined on the xy-plane. It only has the first derivative dy/dx, hence the equation is of the first order. A differential equation of first order can be represented by,
(d/dx) y = f (x,y)
Example:
(dy/dx) + (x2 + 25)y = x/25
Differential equations of First-order and its Types
They are –
Linear Differential Equations
Integrating Factor
hom*ogeneous Equations
Separable Equations
Exact Equations
Characteristics
First-order linear differential equation characteristics are-
It lacks trigonometric and logarithmic functions.
y and its derivatives are absent.
Differential Equation of Second Order
When the maximum derivative is 2, the equation is second order.
Example-
(d2y/dx2) + (x3 + 5x)y = 25
Above equation has the highest derivative is 2. Thus, it is observed to be a second-order differential equation.
Degree
Power of highest order derivative in differential equation represents degree. The differential equation must be a polynomial equation in derivatives to be a differential equation.
For example-
(d4y/dx4) + (d2y/dx2)2 − 3(dy/dx) + y = 9
Here, the degree of the differential equation is 1.
[(d2y/dx2) + (dy/dx)2] = k2(d3y/dx3)2
Here, The order of this equation is 3 and the degree is 2 as the highest derivative is of order 3 and the exponent raised to the highest derivative is 2.
In what circ*mstances does the Degree of Differential Equation have an undefined Value?
Finding the degree of a particular differential equation is not always attainable. If it is written in the form of a polynomial, the degree of the equation can be determined. If the equation is written in another form, the degree cannot be defined.
Assume that the degree of the differential equation dy/dx = sin(x + y) is 1, whereas the degree of the differential equation sin(dy/dx) = x + y is not defined. Differential equations of this type can be seen when other trigonometric functions are used, such as sine, cosine, tan, and so on.
Conclusion
The highest order derivative of the dependent variable with respect to the independent variable is referred to as the order, while the highest degree derivative of the dependent variable is referred to as the degree. The two terms are used to characterise differential equations. In most cases, the differential equation is utilised so as to express a relation that exists between the function and its derivatives. Only when a differential equation is written in the form of a polynomial is it possible to determine the equation’s degree.
