# Determine whether the differential equation is exact. If it is, then solve it. If it is not,...

## Question:

Determine whether the differential equation is exact. If it is, then solve it. If it is not, solve it by first finding the appropriate integrating factor.

(a) {eq}\displaystyle (\sin y - y \sin x)\ dx + (\cos x + x \cos y - y)\ dy = 0 {/eq}.

(b) {eq}\displaystyle 4 x y\ dx + (4 y + 6 x^2)\ dy = 0 {/eq}.

## Exact Differential Equation

An equation of the form M(x,y)dx+N(x,y)dy=0 is exact if

{eq}\frac{{\delta M}}{{\delta y}} = \frac{{\delta N}}{{\delta x}} {/eq}

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a. (sin y-y sin x) dx+(cos x + x cos y-y ) dy=0

Here,

M(x,y)= siny - y sinx

and N(x,y)= cos x + x cos y -y

Now,

{eq}\eqalign{ & \frac{{\delta...