11.1. DEFINITIONS

1. A differential equation is an equation involving differentials or differential coefficients.

   Thus,

   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   

   are all differential equations.

2. Differential equations which involve only one independent variable and the differential coefficients with respect to it are called ordinary differential equations.

   Thus equations (1) to (6) are all ordinary differential equations.

3. Differential equations which involve two or more independent variables and partial derivatives with respect to them are called partial differential equations.

   Thus equations (7) and (8) are partial differential equations.

4. The order of a differential equation is the order of the highest ordered derivative occurring in the differential equation.

   Thus equations (1), (3) and (4) are of the first order ; equations (2) and (6) are of the second order while equation (5) is of the third order.

5. The degree of a differential equation is the degree of the highest ordered derivative which occurs in it provided the equation has been made free of the radical signs and fractional powers as far as the derivatives are concerned.

   Thus, equations (1), (2), (3) and (5) are of the first degree.

   Equation (4) is

   
   

   It is of the second degree.

   Equation (6) is

   It is of the second degree.

6. Solution of a Differential Equation. A solution (or integral) of a differential equation is a relation, free from derivatives, between the variables which satisfies the given equation.

   Thus if y = f( x) is the solution, then by replacing y and...