DIFFERENTIAL EQUATIONS

Differential equation is an equation involving independent and dependent variables and their derivatives (or diferential coefficients). Simply we can say that differential equations means equations which involve differential coefficients.

eg: 1. ( d²y / dx² ) + x ( dy / dx ) + y = 0
2. ( d³y / dx³ ) + x ( d²y / dx² ) + y = 0
3. dy / dx = x cosx
4. ( d²y / dx² ) ² + x² ( dy / dx ) ³ = 0
5. y = ( dy / dx ) + sqrt ( 1 + [dy / dx ] )

The order of a differential equation is the order of the highest order derivative present in the equation.
In the above examples order of equations (1) & (4) are 2 where as equation (3) has 1 & (2) has 3 .

The degree of a differential equation is the degree of the highest order derivative present in the equation, after the equation is free from radicals and fractions.
In the above examples degree of equations (1) ,(2) & (3) is 1 where as equation (4) & (5) has 2 .

A differential equation is said to be linear if it involves dependent variable and its derivatives, occcur in first degree.

The solution of a differential equation is the relation between the independent and dependent variables, not involving its derivatives such that it should satisfy the differential equation.

SOLUTIONS OF FIRST ORDER FIRST DEGREE DIFFERENTIAL EQUATIONS

The general form of a first order first degree differential equation is M dx + N dy =0 where M and N are functions of x and y. Now we procced to develope the solutions of the equations in some particular cases when M and N are certain functions of x and y. So we put forward some methods to solve such differential equations.

1. VARIABLE SEPERABLE FORM
A first order first degree differential equation is said to be in variable seperable form if it is of the form M dx + N dy =0 where M and N are respectively functions of x and y alone.

Problems:
1. Solve ( dy / dx ) = (x² / 1+ y²)