Formation of partial differential equations by elimination of arbitrary functions

FORMATION OF PARTIAL DIFFERENTIAL EQUATIONS BY ELIMINATION OF ARBITRARY FUNCTIONS

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

Solution: Suppose two functions u & v are connected by the relation

Let us eliminate ϕ diff (1) p.w.r. to x and y, we get

which, on simplification gives a p.d.e. of the form

Equation (5) is a first order p.d.e. linear in p and q.

§ What can you say about the order of p.d.e. got by eliminating arbitrary functions?

Solution: The elimination of one arbitrary function will result in a p.d.e. of the first order. The elimination of two arbitrary functions will result in equations of second order and so on.

II. Problems based on Formation of p.d.e. by elimination of arbitrary functions.

Example 1.2(a)(1) : Eliminate f from z = f(x2 + y2).

Solution:

Example 1.2(a)(2) : Eliminate f from z = x + y + f (xy).

Solution:

Example 1.2(a)(3): Eliminate the arbitrary function f from and form a partial differential equation.

Solution: 

Example 1.2(a)(4) : Form the p.d.e. by eliminating ƒ from z = f (x + y).

Solution : 

Example 1.2(a) (5): Eliminate f from z = f(x2 + y2 + z2).

Solution: Given: z = f (x2 + y2 + z2)    ….(1)

⇒ yp + zpq = xq + zpq

yр = xq

i.e., yp – xq = 0 which is the required p.d.e.

Example 1.2(a)(6) : Eliminate f from z = f (xy/z).

Solution: 

Example 1.2(a)(7) : Eliminate ϕ from xyz = ϕ (x + y + z).

Solution:

Differentiating (1) p.w.r. to y, we get

Example 1.2(a) (8): Find the p.d.e. by eliminating the arbitrary function ϕ from ϕ (x2 + y2) = y2 + z2.

Solution:

Example 1.2(a) (9) Form the p.d.e. by eliminating f from

z = xy + f (x2 + y2 + z2).

Solution:

Example 1.2(a)(10) Form the p.d.e. by eliminating ϕ from ϕ (x2 + y2 +z2, x + y + z) = 0.

Solution: Rewriting the given equation as

Example 1.2(a)(11) : Form the p.d.e. by eliminating f from

Solution:

Example 1.2(a)(12): Obtain p.d.e. from z = f (sin x + cos y).

Solution:

Example 1.2(a) (13) Form the p.d.e. by eliminating f from

Solution: 

Example 1.2(a)(14) : Form the PDE by eliminating the arbitrary function from

Solution:

Second Method :

EXERCISE 1.2(a)

Form a p.d.e of eliminating the arbitrary function from

Problems based on Formation of p.d.e by elimination of arbitrary functions

Example 1.2(b)(1) : Form the p.d.e. by eliminating the arbitrary functions f and ϕ from z = f (x + ct) + ϕ (x — ct).

Solution :

Example 1.2(b)(2) : Form the p.d.e. by eliminating f and ϕ from z = f(y) + ϕ (x + y + z).

Solution: 

Differentiating (2) p.w.r. to y, we get

Example 1.2(b)(3) : Eliminate the arbitrary functions f and g from z= f(x + y) + g(x-iy) to obtain a partial differential equation involving z, x, y.

Solution: 

Example 1.2(b)(4): Form the p.d.e. by eliminating f and ϕ from

Solution :

Example 1.2(b)(5): Form the differential equation by eliminating the arbitrary functions ƒ and g in z = f (x3 + 2y) + g (x3 − 2y)

Solution :

which is the required p.d.e.

Example 1.2(b)(6) : Form the partial differential equation by eliminating the arbitrary functions f and g in z = x2f(y) + y2 g(x).

Solution: 

which is the required p.d.e.

Example 1.2(b) (7): Obtain the partial differential equation by eliminating f and g from z = f(2x + y) + g (3x - y).

Solution: 

Example 1.2(b) (8): Form a partial differential equation by eliminating arbitrary functions from z = xf (2x + y) + g (2x + y).

Solution :

Differentiating (3) p.w.r. to x, we get

Differentiating (3) p.w.r. to y, we get

which is the required p.d.e.

EXERCISE 1.2(b).

Form the p.d.e by eliminating the arbitrary functions