# Degree and Order of Differential Equation

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.