An equation that involves independent and dependent variables and at least one derivative of the dependent variable w.r.t independent variable is called a differential equation.


For example: `x dy/dx +y log x = x e^x x^(1/2 log x), (x>0); (d^2y)/(dx^2) = [ y + (dy/dx)^6]^(1/4)`




A differential equation is said to be ordinary, if the differential coefficients have reference to a single independent variable only and it is said to be Partial if there are two or more independent variables. We are concerned with ordinary differential equations only. While an ordinary differential equation containing two or more dependent variables with their differential coefficients w.r.t. to a single independent variable is called a total differential equation.

eg. `(d^2y)/(dx^2) + 3 dy/dx +2y =0` is an ordinary diffrential equation

`(partial u)/(partial x) +(partial u)/(partial y) + (partial u)/(partial z) =0`; `(partial^2 u)/(partial x^2) +(partial^2 u)/(partial y^2)=x^2 +y` are partial differential equation.


`text(Order and Degree of Differential Equation :)`

The order of a differential equation is the order of the highest differential coefficient occuring in it. The degree of a differential equation which is expressed or can be expressed as a polynomial in the derivatives is the degree of the highest order derivative occuring in it, after it has been expressed in a fonn free from radicals & fractions so far as derivatives are concerned, thus the differential equation.

`f(x,y) [ (d^my)/(dx^m)]^p + phi (x,y) [ (d^(m-1) (y))/(dx^(m-1))]^q +..........=0` is order `m` & degree `p` .