Solved Example Problems based on formation of p.d.e by elimination of arbitrary functions

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

(a)  Case (i) a.f. = 1, p.d.e order = 1

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

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

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

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

Example 1.2(a) (3): Eliminate the arbitrary function f from z = f(y/x) and form a partial differential equation.

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

Solution : z = f (x + y)................(1)

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

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

Solution: Given: z = f (x² + ^y2+z²).

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

Solution: Given:

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

Solution: Given: xyz ϕ (x + y + z)…………….(1)

Differentiating (1) p.w.r. to x, 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: Given y²+z² = ᴓ (x²+ y²)

Example 1.2 (a) (9): Form the p.d.e. by eliminating f from z = xy + (x² + y²+z²).

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 

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

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

Example 1.2(a) (14): Form the p.d.e by eliminating the arbitrary function from 

Solution: