Question

Show that the given equation is exact differential equation :

2 2 2 9 (2 1) 0 dy xy x y x dx

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Answer 1

Exact differential equation: A first order and first degree differential equation can be written as

M dx + N dy = 0,

where M and N are functions of x and y.

Now this equation is said to be exact when the differential M dx + N dy will be expressible in the form du without multiplied by any factor, where u is a function of x and y.

The necessary and sufficient condition of exactness: The necessary and sufficient condition that M dx + N dy = 0 be exact is

∂M/∂y = ∂N/∂x

SOLUTION:

We consider a differential equation

x dx + y dy = 0 .... (1)

Comparing (1) with the first order and first degree general equation of differentials, we ge

M = x, N = y

Now ∂M/∂y = ∂/∂y (x) = 0

and ∂N/∂x = ∂/∂x (y) = 0

∵ ∂M/∂y = ∂N/∂x, the differential eqⁿ. (1) is exact.