In the previous section, we completed a discussion on general and particular solution of differential equations. In this section, we will see order and degree of a differential equation.
• First we will see the various notations used to represent derivatives.
(i) First order derivative can be written as:
${\frac{dy}{dx}~~\text{or}~~y'}$
(ii) Second order derivative can be written as:
${\frac{d^2y}{dx^2}~~\rm{or}~~\it{y\,''}}$
(iii) Third order derivative can be written as:
${\frac{d^3y}{dx^3}~~\rm{or}~~\it{y\,'''}}$
(iv) For higher order derivatives, it is inconvenient to use large number of dashes as superscripts. So the nth order derivative can be written as:
${\frac{d^ny}{dx^n}~~\rm{or}~~\it{y_n}}$
Order of a differential equation
•
Order of a differential equation is defined as the order of the highest order derivative in that given differential equation.
•
So the order of a differential equation can be determined in just two steps:
(i) Identify the highest order derivative in the given differential equation.
(ii) The order of that derivative, is the order of the differential equation.
Let us see some examples:
Example 1:
•
The given differential equation is: $\small{\frac{dy}{dx}~=~x^2 + 1}$
•
There is only one derivative. It's order is 1.
•
So the order of the differential equation is 1
Example 2:
•
The given differential equation is: $\small{3\frac{d^2y}{dx^2}~+~4\frac{dy}{dx}-2x^2~=~0}$
•
There are two derivatives: $\small{\frac{d^2y}{dx^2}}$ and $\small{\frac{dy}{dx}}$
•
The highest order derivative is: $\small{\frac{d^2y}{dx^2}}$. It's order is 2.
•
So the order of the differential equation is 2
Example 3:
•
The given differential equation is: $\small{\frac{d^3y}{dx^3}~+~x^2\left(\frac{d^2y}{dx^2} \right)^3~=~0}$
•
There are two derivatives: $\small{\frac{d^3y}{dx^3}}$ and $\small{\frac{d^2y}{dx^2}}$
•
The highest order derivative is: $\small{\frac{d^3y}{dx^3}}$. It's order is 3.
•
So the order of the differential equation is 3.
Degree of a differential equation
• This can be explained in 5 steps:
1. To find the degree of a given differential equation, it is essential that, the differential equation is in the polynomial form.
2. We already know the features of polynomials. Let us recall the main features through some examples:
• ${f(x)=4x^3+5x^2+7}$ is a polynomial function.
• ${f(x)=4x^3+5x^2+\sin(x)}$ is not a polynomial function.
• ${f(x)=4x^3+5\sqrt{x}+7}$ is not a polynomial function.
3. To make such a comparison for differential equations, assume that, in the place of 'x', we have derivatives. So we can write:
• ${4\left(\frac{dy}{dx} \right)^3~+~5\left(\frac{d^2 y}{dx^2} \right)^2+7}$ is a differential equation in the polynomial form.
• ${4\left(\frac{d^3y}{dx^3} \right)^2~+~5\left(\frac{dy}{dx} \right)+\sin x}$ is a differential equation in the polynomial form.
• ${4\left(\frac{dy}{dx} \right)^3~+~\sin\left(\frac{dy}{dx} \right)+7}$ is a differential equation. But it is not in the polynomial form.
• ${4\left(\frac{d^3y}{dx^3} \right)^2~+~\sqrt{\frac{dy}{dx}}+7}$ is a differential equation. But it is not in the polynomial form.
4. If the given differential equation is in the polynomial form, we can write the degree in two steps:
(i) Identify the highest order derivative in the given differential equation.
(ii) The power of that derivative, is the degree of the differential equation.
5. If the given differential equation is not in the polynomial form, we say that, degree of that differential equation is not defined.
Let us see some examples:
Example 1:
• The given differential equation is: $\small{\frac{dy}{dx}~=~x^2 + 1}$
• It is in the polynomial form and hence, degree is defined.
• There is only one derivative.
♦ It's order is 1.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 1.
♦ Degree is 1.
Example 2:
• The given differential equation is: $\small{3\frac{d^2y}{dx^2}~+~4\frac{dy}{dx}-2x^2~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are two derivatives: $\small{\frac{d^2y}{dx^2}}$ and $\small{\frac{dy}{dx}}$
• The highest order derivative is: $\small{\frac{d^2y}{dx^2}}$.
♦ It's order is 2.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 2.
♦ Degree is 1.
Example 3:
• The given differential equation is: $\small{\frac{d^3y}{dx^3}~+~x^2\left(\frac{d^2y}{dx^2} \right)^3~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are two derivatives: $\small{\frac{d^3y}{dx^3}}$ and $\small{\frac{d^2y}{dx^2}}$
• The highest order derivative is: $\small{\frac{d^3y}{dx^3}}$.
♦ It's order is 3.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 3.
♦ Degree is 1.
Example 4:
• The given differential equation is: $\small{\frac{dy}{dx}~=~e^x}$
• It is in the polynomial form and hence, degree is defined.
• There is only one derivative.
♦ It's order is 1.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 1.
♦ Degree is 1.
Example 5:
• The given differential equation is: $\small{\frac{d^2y}{dx^2}~+~y~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There is only one derivative.
♦ It's order is 2.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 2.
♦ Degree is 1.
Example 6:
• The given differential equation is: $\small{\frac{d^3y}{dx^3}~+~x^2\left(\frac{d^2y}{dx^2} \right)^3~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are two derivatives: $\small{\frac{d^3y}{dx^3}}$ and $\small{\frac{d^2y}{dx^2}}$
• The highest order derivative is: $\small{\frac{d^3y}{dx^3}}$.
♦ It's order is 3.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 3.
♦ Degree is 1.
Example 7:
• The given differential equation is: $\small{\frac{d^3y}{dx^3}~+~2\left(\frac{d^2y}{dx^2} \right)^2~-~\frac{dy}{dx}~+~y~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are three derivatives: $\small{\frac{d^3y}{dx^3},~~\frac{d^2y}{dx^2}}$ and $\small{\frac{dy}{dx}}$
• The highest order derivative is: $\small{\frac{d^3y}{dx^3}}$.
♦ It's order is 3.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 3.
♦ Degree is 1.
Example 8:
• The given differential equation is: $\small{\left(\frac{dy}{dx} \right)^2~+~\frac{dy}{dx}~-~\sin^2 y~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There is only one derivative: $\small{\frac{dy}{dx}}$
♦ It's order is 1.
♦ It's highest power is 2.
• So for the differential equation:
♦ Order is 1.
♦ Degree is 2.
Example 9:
• The given differential equation is: $\small{\frac{dy}{dx}~+~\sin\left(\frac{dy}{dx} \right)~=~0}$
• It is not in the polynomial form and hence, degree is not defined.
• There is only one derivative: $\small{\frac{dy}{dx}}$
♦ It's order is 1.
• So for the differential equation:
♦ Order is 1.
We can write two important points:
1. We see that, the order of a differential equation is related to the order of a derivative in that differential equation. We know that, order of any derivative is a positive integer.
• So order of a differential equation will be always a positive integer.
2. We see that, degree (if defined) of a differential equation is related to the power of a derivative. When the degree is defined, the differential equation will be in the polynomial form. In the polynomial form, the powers are all positive integers.
• So degree of a differential equation will be always a positive integer.
Now we will see some solved examples
Solved example 25.14
Find the order of each of the following differential equations. Find also the degree (if defined):
$\small{(i)~~\frac{dy}{dx} - \cos x~=~0}$
$\small{(ii)~~xy \frac{d^2 y}{d x^2} ~+~x \left(\frac{dy}{dx} \right)^2~-~y \frac{dy}{dx}~=~0}$
$\small{(iii)~~y^{'\,'\,'}~+~y^2~+~e^{y'}~=~0}$
Solution:
Part (i):
• The given differential equation is:
$\small{\frac{dy}{dx} - \cos x~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There is only one derivative:
$\small{\frac{dy}{dx}}$
♦ It's order is 1.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 1.
♦ Degree is 1.
Part (ii):
• The given differential equation is:
$\small{xy \frac{d^2 y}{d x^2} ~+~x \left(\frac{dy}{dx} \right)^2~-~y \frac{dy}{dx}~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are two derivatives: $\small{\frac{d^2y}{dx^2}}$ and $\small{\frac{dy}{dx}}$
• The highest order derivative is: $\small{\frac{d^2y}{dx^2}}$.
♦ It's order is 2.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 2.
♦ Degree is 1.
Part (iii):
• The given differential equation is:
$\small{y^{'\,'\,'}~+~y^2~+~e^{y'}~=~0}$
• It is not in the polynomial form and hence, degree is not defined.
• There is only one derivative: $\small{y^{'\,'\,'}}$
♦ It's order is 3.
• So for the differential equation:
♦ Order is 3.
Solved example 25.15
Find the order of each of the following differential equations. Find also the degree (if defined):
$\small{(i)~~\left(\frac{d^2y}{dx^2} \right)^2 + \cos \left(\frac{dy}{dx} \right)~=~0}$
$\small{(ii)~~y^{'\,'\,'} ~+~2y^{'\,'}~+~y' ~=~0}$
Solution:
Part (i):
• The given differential equation is:
$\small{\left(\frac{d^2y}{dx^2} \right)^2 + \cos \left(\frac{dy}{dx} \right)~=~0}$
• It is not in the polynomial form and hence, degree is not defined.
• There are two derivatives: $\small{\frac{d^2y}{dx^2}~~\rm{and}~~\frac{dy}{dx}}$
• The highest order derivative is: $\small{\frac{d^2y}{dx^2}}$.
♦ It's order is 2.
• So for the differential equation:
♦ Order is 2.
Part (ii):
• The given differential equation is:
$\small{y^{'\,'\,'} ~+~2y^{'\,'}~+~y' ~=~0}$
• It is in the polynomial form and hence, degree is defined.
• There are three derivatives: $\small{y^{'\,'\,'},~y^{'\,'}~~\rm{and}~~y'}$
• The highest order derivative is: $\small{y^{'\,'\,'}}$.
♦ It's order is 3.
♦ It's power is 1.
• So for the differential equation:
♦ Order is 3.
♦ Degree is 1.
The links below gives a few more solved examples:
Exercise 9.1
In the next section, we will see formation of differential equations when general solution is given.
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