Differential equations

All the basic concepts you need to know is presented in a very simple way for you to understand.

1. DIFFERENTIAL EQUATIONS An equation involving the Independent Variable x, dependent Variable y and the differential coefficients of dependent Variable with respect to independent variable is called a Differential Equatio n

2. Order of a Differential Equation The Order of a Differential equation is the order of the highest derivative occurring in the Differential equation Eg : (i) + 2 _ = 0 Order of the equation is 3 (ii) = 1 + Order of the equation is 2  

3. Degree of a Differential Equation The Degree of a Differential equation is the degree of the highest derivative occurring in the Differential equation Eg : (i) + 2 _ = 0 Degree of the equation is 1 (ii) = 1 + Degree of the equation is 2 Note : Order and degree (if defined) of a differential equation are always positive integers .  

4. Classifications of Differential Equation Classifications of Differential Equation depends on their (i) Order (ii) Linearity

5. Classifications of Differential Equation according to their Order First Order Differential Equation First Order Differential Equation are those in which only the First Order derivative of the dependent variable occurs. Higher Order Differential Equation Differential equations of order two or more are referred as higher Order Differential Equation

6. Solution of a differential equation A function which satisfies the given differential equation is called its solution.

7. GENERAL AND PARTICULAR SOLUTIONS OF A DIFFERENTIAL EQUATION The solution which contains arbitrary constants is called the general solution ( primitive ) of the differential equation. The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

8. Differential equation to form family of curves   If the given family F1 of curves depends on only one parameter then it is represented by an equation of the form F 1 ( x , y , a ) = 0 If the given family F2 of curves depends on the parameters a , b (say) then it is represented by an equation of the from F 2 ( x , y , a , b ) = 0  

9. DIFFERENTIAL EQUATIONS WITH VARIABLES SEPARABLE If F ( x , y ) can be expressed as a product g ( x ) and h ( y ), where, g ( x ) is a function of x and h ( y ) is a function of y , then the differential equation = F( x,y ) is said to be of variable separable type.  

10. HOMOGENEOUS DIFFERENTIAL EQUATIONS A function F( x , y ) is said to be homogeneous function of degree n if F(  x ,  y ) =  n F( x , y ) for any nonzero constant  . A differential equation which can be expressed in the form = f( x,y ) or = g ( x , y ) where f ( x , y ) and g ( x , y ) are homogenous functions of degree zero is called a homogeneous differential equation  

11. LINEAR DIFFERENTIAL EQUATIONS A differential equation of the form + P y = Q, where P and Q are constants or functions of x only is called a first order linear differential equation.  

12. Classifications of Differential Equation according to their Linearity Linear and non-linear differential equations A differential equation in which the dependent variable and its derivatives occur only in the 1 st degree and are not multiplied together is called a Linear differential equation otherwise it is non-linear .

13. Steps to solve first order linear differential equation (i) Write the given differential equation in the form + P y = Q where P,Q are constants or functions of x only. (ii) Find the Integrating Factor (I.F) = (iii) Write the solution of the given differential equation as y.(I.F) = x I.F) dx +C Note: If the given differential equation is in the form + P 1 x = Q 1 where P 1 ,Q 1 are constants or functions of y only. Then I.F = and the solution of the differential equation is given by x.(I.F ) = x I.F) dy +C