Here you will learn what is differential equation and degree and order of differential equation with examples.
Let’s begin –
Differential Equation
An equation containing an independent variable, dependent variable and differential coefficients of dependent variable with respect to independent variable is called a differential equation.
for example : \(dy\over dx\) = 2xy and \(d^2y\over dx^2\) = 4x are examples of differential equations.
Degree and Order of Differential Equation
Order of Differential Equation
The order of a differential equation is the order of the highest order derivative appearing in the equation.
Example 1 : In the equation \(d^2y\over dx^2\) + 3\(dy\over dx\) + 2y = \(e^x\), the order of highest order derivative is 2. So, it is a differential equation of order 2.
Example 2 : In the equation \(d^3y\over dx^3\) – 6\(({dy\over dx})^2\) – 4y = 0, the order of highest order derivative is 3. So, it is a differential equation of order 3.
Degree of Differential Equation
The degree of a differential equation is the degree of the highest order derivative, when differential coefficients are made free from radicals and fractions.
Example 1 : In the equation \(d^3y\over dx^3\) – 6\(({dy\over dx})^2\) – 4y = 0, the power of highest order derivative is 1. So, it is a differential equation of degree 1.
Example 2 : Consider the differential equation x\(({d^3y\over dx^3})^2\) – 6\(({dy\over dx})^4\) + \(y^4\) = 0.
Solution : In this equation, the order of the highest order derivative is 3 and its power is 2. So, it is a differential equation of order 3 and degree 2.
Example 3 : Consider the differential equation \(({1 + ({dy\over dx})^2})^{3/2}\) = k\({d^2y\over dx^2}\).
Solution : The order of highest order differential coefficient is 2. So, its order is 2.
To find its degree we express the differential equation as a polynomial in derivatives. When expressed as a polynomial in derivatives it becomes \(k^2\)\(({d^2y\over dx^2})^2\) – \(({1 + ({dy\over dx})^2})^3\) = 0. Clearly, the power of the highest order differential coefficient is 2. So, its degree is 2.
