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Differential calculus is the branch of mathematics that studies the rate of change concerning another. It is a subfield of calculus. Equations involving derivatives of a dependent variable with respect to an independent variable are known as differential equations.
Differential equations are classified according to their order. The order of a differential equation (DE) can be defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in that particular differential equation. The degree of a differential equation can be defined as the highest positive integral index power of the highest order of a derivative involved in the given polynomial differential equation. These are the defn. of order and degree of a DE.
The Concept of Differential Calculus and Differentiation
Differential calculus is the branch of mathematics that deals with studying the rate of change of a quantity with respect to another. It is a subfield of calculus.
The primary objective in differential calculus is to study derivatives of a function, differential equations, and their applications in real life.
The function’s derivative describes the rate of change of a function near a chosen input value. The process to find a derivative of a function is termed differentiation.
The derivative of a function at a point is geometrically defined as the area of the slope of a tangent to the function at that given point in a graph.
The process of differentiation is reversed to the process of integration.
The concept of differentiation is applicable in many quantitative processes. It is applied in various concepts of physics, chemistry and mathematics, such as – finding the acceleration of a moving body by differentiating its velocity concerning time; where velocity is given by differentiating the displacement for the time taken by the moving body, in Newton’s second law of motion, finding the rate of a chemical reaction, finding the maxima and minima of a given function, etc.
The Concept of Differential Equations
Equations involving derivatives of a dependent variable with respect to an independent variable are known as differential equations
For example, x (dy / dx) – y = 0 is a differential equation.
An ordinary differential equation is defined as that differential equation that involves the derivatives of a dependent variable with respect to only a single independent variable
For example, (d2y / dx2) + 2 (dy / dx)3 = 0 is an ordinary differential equation.
Differential equations that involve derivatives of a dependent function with respect to more than a single independent variable are known as partial differential equations
For example, (∂2u / ∂t2) – c2 (∂2u / ∂x2) = 0 is a partial differential equation.
Differential equations are classified according to their order
(dy/dx) and (d2y / dx2) are sometimes expressed as y’ and y’’ respectively
Order and Degree of a Differential Equation
We already know that differential equations are classified according to their order. Depending on the order, there can be various differential equation types such as – first-order DE, second-order DE, third-order DE, etc. Here along with the concepts, we will also discuss the defn. of order and degree of a DE.
Order of a Differential Equation
The order of a differential equation (DE) can be defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in that particular differential equation
The order of a differential equation is also known as the differential coefficient of that function
For example, the order of the differential equations:
(d2y / dx2) + 2 (dy / dx)3 = 0 is 2.
x (dy / dx) – y = 0 is 1.
(d3y / dx3) + x3 (d2y / dx2) + 3 (dy / dx)3 = 0 is 3.
Differential equations involving first-order derivatives and second-order derivatives are known as first-order DE and second-order DE, respectively
It is similar to other orders of a DE
Degree of a Differential Equation
The degree of a differential equation can be defined as the highest positive integral index power of the highest order of a derivative involved in the given polynomial differential equation
It is to be noted that the degree of a differential equation can only be found in the given differential equation is a polynomial
For example, the degree of the given polynomial differential equations is:
(dy/dx) – ex = 0 is 1.
(d3y/dx3) + x3 (d2y/dx2) + 3 (dy/dx)3 = 0 is 1.
(dy/dx)2 + 2 (dy/dx) + cos2y = 0 is 2.
Conclusion
Through this article on Calculus-Differential Equations-Defn. of Order and Degree of a DE – Maths, students have been made familiar with the concepts of differential calculus and differentiation, differential equations, order of a differential equation, the basis of classification of differential equations, and their degree.
They now know that the order of a differential equation (DE) can be defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in that particular differential equation. The degree of a differential equation can be defined as the highest positive exponential power of the highest order of a derivative involved in the given polynomial differential equation.
