Order and Degree of Differential Equations

Equations having independent variable, dependent variable and their derivatives or derivative (differential) coefficients of one variable with respect to the other variable are known as differential equations — for example, y=2x2.  

Differentiating with respect to x,

 dydx=2.2x 

 dydx=4x

This is a differential equation as it contains an independent variable (x), a dependent variable (y) and its derivative. Considering the order and degree of the differential equation example taken above, we can see that it has been differentiated only once. Here, dydx is the highest order differential. Hence, the order of this differential equation is one, and the largest power (degree) of this highest order derivative is one, so the degree of the given differential equation is one. Therefore, the order of the differential equation is the highest order differential (derivative) we have in the equation. The largest power (exponent or degree) of the highest order differential (derivative) present is the degree of the differential equation.

Order of the differential equation

Consider the differential equations given below.

dydx+3ex=0

d2ydx2+y=0

d4ydx4+sin d3ydx3 =0

The order of the first equation seems to be one, the order of the second equation is two, and that of the third equation is four. This is because the order of any differential equation considered is always the highest order derivative present in the equation of the dependent variable with respect to the independent variable engaged in that differential equation. Differential equations can be differentiated based on their order as the first order, second order and so on, till the nth order differential equation.

First-order differential equation

From the above examples, considering example 1, 

dydx+3ex=0

We can see that its highest order derivative is one, and hence its order is one. So, a differential equation with the highest degree one is a first-order differential equation. The linear equations in the form of differentials are also first-order differential equations.

Second-order differential equation

Considering the second example, we have taken, 

d2ydx2+y=0

We can see that the highest order derivative present is two, and hence its order is one. So, an equation having the highest order derivative of order two is considered a second-order differential equation.

Some differential equations of different orders

d3ydx3+sin d2ydx2 =0 is a third-order differential equation.

dnydxn+Aydn-1ydxn-1+…+Zyn=K is an nth-order differential equation.

Degree of the differential equation

The highest power (exponent) of the highest order derivative of any differential equation is its degree. For example,

(dydx)2+dydx-x=0

d2ydx2+dydx+sin dydx+1=0

The highest order derivative in the first example is of order one, and its highest exponent is two. Hence, the degree of the first example is two, but for the second example, the degree is not defined. So, some differential equations have no degree. For the polynomial equations not in their derivative form (in which the polynomial in dependent variable exists), the degree of such equations doesn’t exist.

Degree (if exists) always exists in positive integer form. If the equation is given as d2ydx2=dydx+1, to determine its degree, we first have to square on both sides to make the exponents in integer form; only then can we determine the degree of such equations. For this one, after squaring on both sides, we get

d2ydx22=dydx+1.

Now, the exponent of the highest order derivative (differential) is a positive integer, i.e., two. Hence, the degree of this differential equation is two.

Linear differential equation

For a differential equation to be a linear differential equation, there should be no multiplication of two different order derivatives. Or there should be a multiplication of the dependent variable and the differential (derivative) present. Or the derivative and dependent variable present in the equation should be of degree one. If an equation doesn’t satisfy any of the criteria mentioned above, it is considered a nonlinear differential equation.

First-order first-degree differential equations

An equation involving only the first order derivative of the dependent variable, the dependent variable itself and an independent variable is considered as first order first degree differential equations. The general form can be written as 

Fx,y,y‘=0

Conclusion

Order of any differential equation is the highest order differential present, of the dependent variable with respect to the independent variable in the differential equation. It can be found by first calculating the derivatives of the given equation. Degree of any differential equation is the highest power (exponent) of the highest order derivative present in the equation. Both order and degree (if present) of a differential equation exists in positive integer form. With the help of order and degree of the differential equations, we can identify their type and the complexity of the equations given. Order and degree help in further solution of differential equations and in several other related topics also.

Here are some questions on order and degree of differential equations for better understanding.