`=>` In general, an equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.

`=>` A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation, e.g.,

`color{orange} {2 (d^y)/(dx^2) + ((dy)/(dx) )^3 = 0}` is an ordinary differential equation.


Note : 1. . We shall prefer to use the following notations for derivatives:

`color{blue}{((dy)/(dx)) = y' ;\ \ ((d^2y)/(dx^2)) = y'' ;\ \((d^3y)/(dx^3)) = y''' }`

2. For derivatives of higher order, it will be inconvenient to use so many dashes as supersuffix therefore, we use the notation `y_n`

for `n^(th)` order derivative `color{blue}{((d^n y)/(dx^n)).}`

`=>` Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

Consider the following differential equations:

`color{orange} {(dy)/(dx) = e^x}` .......(6)

`color{orange} {(d^2y)/(dx^2) + y = 0}`.........(7)

`color{orange} {((d^3y)/(dx^3) ) + x^2 ((d^2y)/(dx^2))^3 = 0}` .........(8)

`=>` The equations (6), (7) and (8) involve the highest derivative of first, second and third order respectively.

`=>` To study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc.

Consider the following differential equations:

`color{orange} {(d^3y)/(dx^3) + 2 ((d^2y)/(dx^2))^2 - (dy)/(dx) + y = 0}` .......(9)

`color{orange} {((dy)/(dx))^2 + ((dy)/(dx)) - sin^2 y = 0}`.......(10)

`color{orange} {(dy)/(dx) + sin ((dy)/(dx)) = 0}` .........(11)

`=>` By the degree of a differential equation, when it is a polynomial equation in derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.

`=>` In view of the above definition, one may observe that differential equations (6), (7), (8) and (9) each are of degree one, equation (10) is of degree two while the degree of differential equation (11) is not defined.

Note : Order and degree (if defined) of a differential equation are always positive integers.