Formation of partial differential equations by elimination of arbitrary Constants

FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS BY ELIMINATION OF ARBITRARY CONSTANTS.

Consider an equation f (x, y, z, a, b) = 0 .... (1)

where a and b denote arbitrary constants.

Let z be regarded as function of two independent variables x and y.

Differentiating (1) with respect to x and y partially, we get

Eliminating two constants a and b from three equations, we shall obtain an equation of the form φ (x, y, z, p, q) = 0

which is partial differential equation of the first order.

Note 1: In a similar manner, it can be shown that if there are more arbitrary constants than the number of independent variables, the above procedure of elimination will give rise to partial differential equations of higher order than the first.

Note 2: f(x, y, z, a, b) = 0 is called the complete solution of y (x, y, z, p, q)

§ Define 'a partial differential equation'.

Solution : A p.d.e is one which involves partial derivatives for instance 

 are all partial differential equations.

§ Define the order of a p.d.e and its degree.

Solution: The order of a p.d.e is the order of the highest partial differential coefficient occurring in it.

The degree of the highest derivative is the degree of the p.d.e.

§ When is a p.d.e said to be linear?

Solution: A p.d.e is said to be linear, if the dependent variable and the partial derivatives occur in the first degree only and separately.

§ Distinguish between homogeneous and non-homogeneous p.d.e.

Solution: An equation of the type

is called a homogeneous linear p.d.e. of nth order with constant co-efficients. It is called homogeneous because all the terms contain derivatives of the same order.

The linear differential equations which are not homogeneous, are called non-homogeneous linear equations.

§ Explain how p.d.e. is formed.

Solution: P.d.e. can be obtained

(i) by eliminating the arbitrary constants that occur in the functional relation between the dependent and independent variables. (or)

(ii) by eliminating arbitrary functions from a given relation between the dependent and independent variables.

§ What is the essential difference between ordinary differential equation and p.d.e., when both are formed by eliminating arbitrary constants ?

Solution: The order of an ordinary differential equation will be the same, as the number of constants eliminated. The order of a p.d.e. will be one, in cases, when the number of constants to be eliminated is equal to the number of independent variables. But if the number of constants to be eliminated is more than the number of independent variables, the result of the elimination will, in general, be p.d.e. of second and higher orders.

§ Explain the formation of p.d.e. by elimination of arbitrary constants.

Solution: A p.d.e. is formed by eliminating the arbitrary constants that occur in the functional relation between the variables.

Let f (x, y, z, a,b) = 0 (1) be a relation

Connecting x, y, z and the arbitrary constants a, b, in (1), z is considered as the dependent variable.