An equation that involves an independent variable , dependent variable and differential coefficients of dependent variable with respect to the independent variable is called a differential equation .
e . g ., ( i ) x 2 ( d 2 y / dx 2 ) + x 3 ( dy / dx ) 3 7x 2 y 2
( ii ) ( x 2 + y 2 ) dx = ( x 2 – y 2 ) dy
Order and Degree of a Differential Equation
The order of a differential equation is the order of the highest derivative occurring in the equation . The order of a differential equation is always a positive integer .
The degree of a differential equation is the degree ( exponent ) of the derivative of the highest order in the equation , after the equation is free from negative and fractional powers of the derivatives .
Linear and Non-Linear Differential Equations
A differential equation is said to be linear , if the dependent variable and all of its derivatives occurring in the first power and there are no product of these . A linear equation of nth order can be written in the form
where , P0 , P1 , P2 ,…, Pn – 1 and Q must be either constants or functions of x only .
A linear differential equation is always of the first degree but every differential equation of the first degree need not be linear .
e . g ., The equations d 2 y / dx 2 + ( dy / dx ) 2 + xy = 0 and
x ( d 2 y / dx 2 ) + y ( dy / dx ) + y = x 3 , ( dy / dx ) d 2 y / dx 2 + y = 0
are not linear .